∎

¹¹institutetext: Abolfazl Asudeh ²²institutetext: Department of Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
²²email: [email protected] ³³institutetext: Tanya Berger-Wolf ⁴⁴institutetext: Department of Computer Science and Engineering, Ohio State University, Columbus, OH 43210, USA
⁴⁴email: [email protected] ⁵⁵institutetext: ^∗Bhaskar DasGupta (corresponding author) ⁶⁶institutetext: Department of Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
Tel.: +312-355-1319
Fax: +312-413-0024
⁶⁶email: [email protected] ⁷⁷institutetext: Anastasios Sidiropoulos ⁸⁸institutetext: Department of Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
⁸⁸email: [email protected]

Maximizing coverage while ensuring fairness: a tale of conflicting objectives

Abolfazl Asudeh Tanya Berger-Wolf Bhaskar DasGupta^∗ Anastasios Sidiropoulos

(Received: date / Accepted: date)

Abstract

Ensuring fairness in computational problems has emerged as a key topic during recent years, buoyed by considerations for equitable resource distributions and social justice. It is possible to incorporate fairness in computational problems from several perspectives, such as using optimization, game-theoretic or machine learning frameworks. In this paper we address the problem of incorporation of fairness from a combinatorial optimization perspective. We formulate a combinatorial optimization framework, suitable for analysis by researchers in approximation algorithms and related areas, that incorporates fairness in maximum coverage problems as an interplay between two conflicting objectives. Fairness is imposed in coverage by using coloring constraints that minimizes the discrepancies between number of elements of different colors covered by selected sets; this is in contrast to the usual discrepancy minimization problems studied extensively in the literature where (usually two) colors are not given a priori but need to be selected to minimize the maximum color discrepancy of each individual set. Our main results are a set of randomized and deterministic approximation algorithms that attempts to simultaneously approximate both fairness and coverage in this framework.

Keywords:

fairness maximum coverage combinatorial optimization approximation algorithms

^†^†journal:

1 Introduction

In this paper we introduce and analyze a combinatorial optimization framework capturing two conflicting objectives: optimize the main objective while trying to ensure that the selected solution is as fair as possible. We illustrate the framework with the following simple graph-theoretic illustration. Consider the graph $G$ of $10$ nodes and $18$ edges as shown in Fig. 1 where each edge is colored from one of $\chi=3$ colors (red, blue or green) representing three different attributes. Suppose that we want to select exactly $k=3$ nodes that maximizes the number of edges they “cover” subject to the “fairness” constraint that the proportion of red, blue and green edges in the selected edges are the same. An optimal solution is shown in Fig. 1 by the solid black nodes $u_{1},u_{2},u_{3}$ covering $6$ edges; Fig. 1 also shows that the solution is quite different from what it would have been (the yellow corner nodes $v_{1},v_{2},v_{3}$ covering $11$ edges) if the fairness constraint was absent. A simple consequence of the analysis of our algorithms for a more general setting is that, assuming that there exists at least one feasible solution and assuming $k$ is large enough, we can find a randomized solution to this fair coverage problem for graphs where we select exactly $k$ nodes, cover at least $63\%$ of the optimal number of edges on an average and, for every pair of colors, with high probability the ratio of the number of edges of these two colors among the selected edges is $O(1)$ .

Refer to caption — Figure 1: A simple illustration of fairness in maximum coverage problems for graphs.

In this paper we consider this type of problem in more general settings. Of course, in the example in Fig. 1 (and in general) there is nothing special about requiring that the proportion of red, blue and green edges in the covered edges should be exactly equal as opposed to a pre-specified unequal proportion. For example, we may also require that the proportion of edges of different colors in our solution should mimic that in the entire graph, i.e., in Fig. 1 among the covered edges the proportion of red, blue and green edges should be $q_{1},q_{2}$ and $q_{3}$ where $q_{1}=\nicefrac{{1}}{{6}}$ , $q_{2}=\nicefrac{{1}}{{3}}$ , and $q_{3}=\nicefrac{{1}}{{2}}$ . Our algorithms will work with easy modifications for any constant values of $q_{1}$ , $q_{2}$ and $q_{3}$ .

1.1 Different research perspectives in ensuring fairness

Theoretical investigations of ensuring fairness in computation can be pursued from many perspectives. We briefly comment on a few of them.

One line of research dealing with the goal of ensuring fairness uses the optimization framework, i.e., we model the problem as an optimization problem with precisely defined fairness constraints. This is a common framework used by researchers in combinatorial and graph-theoretic algorithms, such as research works that involve designing exact or approximation algorithms, investigating fixed-parameter tractability issues or proving inapproximability results. In this paper we use such a framework. Fairness is imposed in coverage by using coloring constraints that minimizes the discrepancies between different colors among elements covered by selected sets; this is in contrast to the usual discrepancy minimization problems studied extensively in the literature C00 where the (usually two) colors are not given a priori but need to be selected to minimize the maximum color discrepancy of each individual set.

A second line of research dealing with fairness involves machine learning frameworks. Even though it is a relatively new research area, there is already a large body of research dealing with ensuring fairness in machine learning algorithms by preprocessing the data used in the algorithms, optimization of statistical outcomes with appropriate fairness criteria and metric during the training, or by post-processing the answers of the algorithms Ze13 ; HPS16 ; ZLM18 .

A third line of research dealing with fairness involves game theoretic frameworks. For example, developments of solutions for fair ways of sharing transferable utilities in cooperative game-theoretic environments have given rise to interesting concepts such as Shapley values and Rabin’s fairness model. We refer the reader to the excellent textbook in algorithmic game theory by Nisan et al. NRTV07 for further details on these research topics.

Yet another more recent line of research dealing with fairness involve applying fairness criteria in the context of clustering of points in a metric space under $k$ -means objective, $k$ -median or other $\ell_{p}$ -norm objectives BCFN19 ; CKLV17 . The assumption of an underlying metric space allows the development of efficient algorithms for these frameworks.

2 Fair maximum coverage: notations, definitions and related concepts

The Fair Maximum Coverage problem with $\chi$ colors is defined as follows. We are given an universe $\mathcal{U}=\{u_{1},\dots,u_{n}\}$ of $n$ elements, a weight function $w:\mathcal{U}\mapsto{\mathbb{R}}$ assigning a non-negative weight to every element, a color function $\mathcal{C}:\mathcal{U}\mapsto\{1,\dots,\chi\}$ assigning a color to every element, a collection of $m$ sets $\mathcal{S}_{1},\dots,\mathcal{S}_{m}\subseteq\mathcal{U}$ , and a positive integer $k$ . A collection of $k$ distinct subsets, say $\mathcal{S}_{i_{1}},\dots,\mathcal{S}_{i_{k}}$ , with the set of “covered” elements $\bigcup_{j=1}^{k}\mathcal{S}_{i_{j}}$ containing $p_{i}$ elements of color $i$ is considered a valid solution¹¹1For a more general version of the problem we are given $\chi$ “color-proportionality constants” $q_{1},\dots,q_{\chi}\in(0,1]$ with $q_{1}+\dots+q_{\chi}=1$ , and a valid solution must satisfy $\nicefrac{{p_{i}}}{{p_{j}}}=\nicefrac{{q_{i}}}{{q_{j}}}$ for all $i$ and $j$ . As we mentioned already, with suitable modifications our algorithms will work with similar asymptotic performance guarantee for any constant values of $q_{1},\dots,q_{\chi}$ , but to simplify exposition we will assume the simple requirement of $q_{1}=\dots=q_{\chi}$ in the sequel. provided $p_{i}=p_{j}$ for all $i$ and $j$ . The objective is to maximize the sum of weights of the covered elements. More explicitly, our problem is defined as follows:

Problem name:

Fair Maximum Coverage with

\chi

colors (Fmc(

\chi,k

))

Input:

\bullet

universe

\mathcal{U}=\{u_{1},\dots,u_{n}\}

\bullet

(element) weight function

w:\mathcal{U}\mapsto{\mathbb{R}}^{+}\cup\{0\}

\bullet

(element) color function

\mathcal{C}:\mathcal{U}\mapsto\{1,\dots,\chi\}

\bullet

sets

\mathcal{S}_{1},\dots,\mathcal{S}_{m}\subseteq\mathcal{U}

\bullet

integer

k>0

Valid solution:

collection of

k

distinct subsets

\mathcal{S}_{i_{1}},\dots,\mathcal{S}_{i_{k}}

satsifying

\forall i,j\in\{1,\dots,\chi\}:

p_{i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\big{|}\,\{u_{\ell}\,|\,u_{\ell}\in\bigcup_{j=1}^{k}\mathcal{S}_{i_{j}}\text{ and }\mathcal{C}(u_{\ell})=i\}\,\big{|}

=

p_{j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\big{|}\,\{u_{\ell}\,|\,u_{\ell}\in\bigcup_{j=1}^{k}\mathcal{S}_{i_{j}}\text{ and }\mathcal{C}(u_{\ell})=j\}\,\big{|}

Objective:

maximize

\sum_{u_{\ell}\in\bigcup_{j=1}^{k}\mathcal{S}_{i_{j}}}w(u_{\ell})

We denote Fmc( $\chi,k$ ) by just Fmc when $\chi$ and $k$ are clear from the context. In the sequel, we will distinguish between the following two versions of the problem:

(i)

unweighted Fmc in which $w(u_{\ell})=1$ for all $\ell\in\{1,\dots,n\}$ and thus the objective is to maximize the number of elements covered, and
(ii)

weighted Fmc in which $w(u_{\ell})\geq 0$ for all $\ell\in\{1,\dots,n\}$ .

For the purpose of stating and analyzing algorithmic performances, we define the following notations and natural parameters associated with an instance of Fmc( $\chi,k$ ):

$\triangleright$

$a\in\{2,3,\dots,n\}$ denotes the maximum of the cardinalities (number of elements) of all sets.
$\triangleright$

$f\in\{1,2,\dots,m\}$ denotes the maximum of the frequencies of all elements, where the frequency of an element is the number of sets in which it belongs.
$\triangleright$

${\mathsf{OPT}}$ denotes the optimal objective value of the given instance of Fmc.
$\triangleright$

${\mathsf{OPT}}_{\#}$ denotes the number of covered elements in an optimal solution of the given instance of Fmc. For weighted Fmc, if there are multiple optimal solutions then ${\mathsf{OPT}}_{\#}$ will the maximum number of elements covered among these optimal solutions. Note that ${\mathsf{OPT}}={\mathsf{OPT}}_{\#}$ for unweighted Fmc. The reason we need to consider ${\mathsf{OPT}}_{\#}$ separately from ${\mathsf{OPT}}$ for weighted Fmc is because the coloring constraints are tied to ${\mathsf{OPT}}_{\#}$ whereas the optimization objective is tied to ${\mathsf{OPT}}$ .
$\triangleright$

The performance ratios of many of our algorithms are expressed using the function $\varrho(\cdot)$ :

${\varrho(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}{\big{(}1-\nicefrac{{1}}{{x}}\big{)}}^{x}}$

Note that $\varrho(x)<\varrho(y)$ for $x>y>0$ and $\varrho(x)>1-{\mathbf{e}}^{-1}$ for all $x>0$ .

For $\mathsf{NP}$ -completeness results, if the problem is trivially in $\mathsf{NP}$ then we will not mention it. To analyze our algorithms in this paper, we have used several standard mathematical equalities or inequalities which are listed explicitly below for the convenience of the reader:

	$\displaystyle\forall\,x\in[0,1]\,:\,{\mathbf{e}}^{-x}\geq 1-x$		(1)
	$\displaystyle\forall\,x\,:\,{\mathbf{e}}^{-x}=1-x+({x^{2}}/{2}){\mathbf{e}}^{-\xi}\text{ for some }\xi\in[0,x]$		(2)
	$\displaystyle\textstyle\forall\,\alpha_{1},\dots,\alpha_{q}\geq 0\,:\,\left(\frac{1}{q}{\sum_{j=1}^{q}\alpha_{j}}\right)^{\!q}\geq\prod_{j=1}^{q}\alpha_{j}$		(3)
	$\displaystyle\textstyle\forall\,x\in[0,1]\,\,\forall\,y\geq 1\,:\,1-\left(1-\frac{x}{y}\right)^{y}\geq\left(1-\left(1-\frac{1}{y}\right)^{y}\right)x$		(4)

2.1 Three special cases of the general version of Fmc

In this subsection we state three important special cases of the general framework of Fmc.

Fair maximum $k$ -node coverage or Node-fmc

This captures the scenario posed by the example in Fig. 1. We are given a connected undirected edge-weighted graph $G=(V,E)$ where $w(e)\geq 0$ denotes the weight assigned to edge $e\in E$ , a color function $\mathcal{C}:E\mapsto\{1,\dots,\chi\}$ assigning a color to every edge, and a positive integer $k$ . A node $v$ is said to cover an edge $e$ if $e$ is incident on $v$ . A collection of $k$ nodes $v_{i_{1}},\dots,v_{i_{k}}$ covering $p_{i}$ edges of color $i$ for each $i$ is considered a valid solution provided $p_{i}=p_{j}$ for all $i$ and $j$ . The objective is to maximize the sum of weights of the covered edges. It can be easily seen that this is a special case of Fmc by using the standard translation from node cover to set cover, i.e., the edges are the set of elements, and corresponding to every node $v$ there is a set containing the edges incident on $v$ . Note that for this special case $f=2$ and $a$ is equal to the maximum node-degree in the graph.

Segregated Fmc or Segr-fmc

Segregated Fmc is the special case of Fmc when all the elements in any set have the same color, i.e.,

\forall\,j\in\{1,\dots,m\}\,\forall\,u_{p},u_{q}\in\mathcal{S}_{j}\,:\,\mathcal{C}(u_{p})=\mathcal{C}(u_{q})

Another equivalent way of describing Segr-fmc is as follows. Let $C_{j}$ is the set of all elements colored $j$ for $j\in\{1,\dotsc,\chi\}$ in the given instance of Segr-fmc. Let the notation $2^{A}$ denote the power set for any set $A$ . Then, Segr-fmc is the special case when $\mathcal{S}_{j}\in\bigcup_{j=1}^{\chi}2^{C_{j}}$ holds for all $j\in\{1,\dots,m\}$ . A simple example of an instance of Segr-fmc with $n=6$ , $m=10$ and $\chi=2$ is shown below:

	$\displaystyle\mathcal{U}=\{u_{1},u_{2},u_{3},u_{4},u_{5},u_{6}\}$
	$\displaystyle\mathcal{C}(u_{1})=\mathcal{C}(u_{2})=\mathcal{C}(u_{3})=\mathcal{C}(u_{4})=1,\,\,\,\mathcal{C}(u_{5})=\mathcal{C}(u_{6}))=2$
	$\displaystyle\mathcal{S}_{1}=\{u_{1},u_{2},u_{3}\},\,\,\,\mathcal{S}_{2}=\{u_{2},u_{3},u_{4}\},\,\,\,\mathcal{S}_{3}=\{u_{5},u_{6}\},\,\,\,\mathcal{S}_{4}=\{u_{6}\}$
	$\displaystyle w(u_{1})=w(u_{2})=7,,\,\,\,w(u_{3})=9,,\,\,\,w(u_{4})=w(u_{5})=w(u_{6})=1$

Even though computing an exact solution of Segr-fmc is still $\mathsf{NP}$ -complete, it is much easier to approximate (see Section 11.1). From our application point of view as discussed in Section 3, this may for example model cases in which city neighborhoods are segregated in some manner, e.g., racially or based on income.

$\Delta$ -balanced Fmc or $\Delta$ -bal-fmc

$\Delta$ -balanced Fmc is the special case of Fmc when the number of elements of each color in a set are within an additive range of $\Delta$ , i.e.,

\forall\,j\in\{1,\dots,m\}\,\forall\,p\in\{1,\dots,\chi\}\,:\,\\ \max\left\{1,\,\left\lfloor\frac{|\mathcal{S}_{j}|}{\chi}\right\rfloor-\Delta\right\}\leq\left|u_{\ell}\,|\,(u_{\ell}\in\mathcal{S}_{j})\wedge(\mathcal{C}(u_{\ell})=p)\right|\leq\left\lceil\frac{|\mathcal{S}_{j}|}{\chi}\right\rceil+\Delta

Similar to Segr-fmc, it is much easier to approximate $\Delta$ -bal-fmc for small $\Delta$ (see Section 11.2).

Geometric Fmc or Geom-fmc

In this unweighted “geometric” version of Fmc, the elements are points in $[0,\Delta]^{d}$ for some $\Delta$ and some constant $d\geq 2$ , the sets are unit radius balls in ${\mathbb{R}}^{d}$ , and the distributions of points of different colors are given by $\chi$ Lipschitz-bounded measures. More precisely, the distribution of points of color $i$ is given by a probability measure $\mu_{i}$ supported on $[0,\Delta]^{d}$ with a $C$ -Lipschitz density function²²2A function $f:\Delta\mapsto{\mathbb{R}}$ for some subset $\Delta$ of real numbers is $C$ -Lipschitz provided $|f(x)-f(y)|\leq C\,|x-y|$ for all real numbers $x,y\in\Delta$ . for some $C>0$ that is upper-bounded by $1$ . Given a set of $k$ unit balls $\mathcal{B}_{1},\dots,\mathcal{B}_{k}\subset{\mathbb{R}}^{d}$ , the number of points $p_{i}$ of color $i$ covered by these balls is given by $\mu_{i}\big{(}\bigcup_{i=1}^{k}\mathcal{B}_{i}\big{)}$ , and the total number of points covered by these balls is given by $\sum_{i=1}^{\chi}\mu_{i}\big{(}\bigcup_{i=1}^{k}\mathcal{B}_{i}\big{)}$ . This variant has an almost optimal polynomial-time approximation algorithm for fixed $d$ and under some mild assumption on ${\mathsf{OPT}}$ (see Section 12).

3 Sketch of application scenarios

Fmc and its variants are core abstractions of many data-driven societal domain applications. We present three diverse categories of application and highlight the real-world fairness issues addressed by our problem formulations (leaving other applications in the cited references).

Service/Facility Allocation

One of the most common data based policy decisions is assigning services/facilities across different places, e.g., placing schools ChicagoSchoolClosure , bus stops, or police/fire stations, choosing a few hospitals for specific medical facilities or services, or deciding where to put cell-phone towers. Of course, a major objective in such assignments is to serve the maximum number of people (i.e., maximize the coverage). Unfortunately, historical discriminations, such as redlining redlining , through their long drawn-out effects of manifestations in different aspects of public policy are still hurting the minorities. As a result, blindly optimizing for maximum coverage biases the assignment against equitable distribution of services. Below are examples of two real cases that further underline the importance of fairness while maximizing coverage:

$\triangleright$

Bike sharing: As more and more cities adopt advanced transportation systems such as bike-sharing, concerns such as equity and fairness arise with them yan2019fairness . For instance, according to BikeShare1 the bike-sharing network at NYC neglects many low-income neighborhoods and communities of color while giving the priority to well-to-do neighborhoods. Here the location of bike stations (or bikes) determines the set of people that will have access to the service, perpetuating the unhealthy cycle of lack of transportation, movement, etc.
$\triangleright$

Delivery services for online shopping: Online shopping has by now gained a major share of the shopping market. Platforms such as Amazon provide services such as same-day delivery to make e-shopping even more convenient to their customers. While Amazon’s main aim is to maximize the number of customers covered by this service, by not considering fairness it demonstrably failed to provide such services for predominantly black communities AmazonDeliveryBias1 ; AmazonDeliveryBias2 .

Data Integration

Combining multiple data sources to augment the power of any individual data source is a popular method for data collection. Naturally the main objective of data integration is to collect (“cover”) a maximum number of data points. However, failing to include an adequate number of instances from minorities, known as population bias, in datasets used for training machine learning models is a major reason for model unfairness olteanu2019social ; asudeh2019assessing . For example, image recognition and motion detection services by Google google-gorilla and HP hp1 with a reasonable overall performance failed to tag/detect African Americans since their training datasets did not include enough instances from this minority group. While solely optimizing for coverage may result in biased datasets, considering fairness for integration may help remove population bias.

Targeted advertisement

Targeted advertising is popular in social media. Consider a company that wants to target its “potential customers”. To do so, the company needs to select a set of features (such as “single” or “college student”) that specify the groups of users to be targeted. Of course, the company wants to maximize coverage over its customers. However, solely optimizing for coverage may result in incidents such as racism in the Facebook advertisements AdRacism or sexism in the job advertisements adSexism . Thus, a desirable goal for the company would be to select the keywords such that it provides fair coverage over users of diverse demographic groups.

4 Review of prior related works

To the best of our knowledge, Fmc in its full generalities has not been separately investigated before. However, there are several prior lines of research that conceptually intersect with Fmc.

Maximum $k$ -set coverage and $k$ -node coverage problems

The maximum $k$ -set coverage and $k$ -node coverage problems are the same as the Fmc and Node-fmc problems, respectively, without element colors and without coloring constraints. These problems have been extensively studied in the algorithmic literature, e.g., see AS04 ; F98 ; Ho97 for $k$ -set coverage and GLL18a ; GLL18b ; GNW07 ; M18 ; FL01 ; AS14 ; HYZZ02 for $k$ -node coverage. A summary of these results are as follows:

$k$ -set coverage:: The best approximation algorithm for $k$ -set coverage is a deterministic algorithm that has an approximation ratio of $\max\big{\{}\varrho(f),\,\varrho(k)\big{\}}>1-\nicefrac{{1}}{{{\mathbf{e}}}}$ AS04 ; Ho97 . On the inapproximability side, assuming $\mathsf{P}\!\neq\!\mathsf{NP}$ an asymptotically optimal inapproximability ratio of $1-\nicefrac{{1}}{{{\mathbf{e}}}}+\varepsilon$ (for any $\varepsilon>0$ ) is known for any polynomial-time algorithm F98 .
$k$ -node coverage:: The best approximation for $k$ -node coverage is a randomized algorithm that has an approximation ratio of $0.7504$ with high probability FL01 ; HYZZ02 . On the inapproximability side, $k$ -node coverage is $\mathsf{NP}$ -complete even for bipartite graphs AS14 , and cannot be approximated within a ratio of $1-\varepsilon$ for some (small) constant $\varepsilon>0$ L98 ; P94 . More recently, Manurangsi M18 provided an semidefinite programming based approximation algorithm with an approximation ratio of $0.92$ , and Austrin and Stankovic AS19 used the results in AKS11 to provide an almost matching upper bound of $0.929+\varepsilon$ (for any $\varepsilon>0$ ) on the approximation ratio of any polynomial time algorithm assuming the unique games conjecture is true. There is also a significant body of prior research on the fixed parameter tractability issues for the $k$ -node coverage problem: for example, $k$ -node coverage is unlikely to allow an FPT algorithm as it is $W[1]$ -hard GNW07 , but Marx designed an FPT approximation scheme in M08 whose running times were subsequently improved in Gupta, Lee and Li in GLL18a ; GLL18b .

However, the coloring constraints make Fmc fundamentally different from the maximum set or node coverage problems. Below we point out some of the significant aspects of these differences. For comparison purposes, for an instance of Fmc let ${\mathsf{OPT}}_{\mathrm{coverage}}$ denote the objective value of an optimal solution for the corresponding maximum $k$ -set coverage problem for this instance by ignoring element colors and coloring constraints.

Existence of a feasible solution:: For the maximum $k$ -set coverage problem, a feasible solution trivially exists for any $k$ . However, a valid solution for Fmc( $\chi,k$ ) may not exist for some or all $k$ even if $\chi=2$ and in fact our results (Lemma 1) show that even deciding if there exists a valid solution is $\mathsf{NP}$ -complete. The $\mathsf{NP}$ -completeness result holds even if $f=1$ (i.e., the sets are mutually disjoint); note that if $f=1$ then it is trivial to compute an optimal solution to the maximum $k$ -set coverage problem. That is why for algorithmic purposes we will assume the existence of at least one feasible solution³³3Actually, our ${\mathsf{LP}}$ -relaxation based algorithms require only the existence of a feasible fractional solution but we cannot say anything about the approximation ratio in the absence of a feasible integral solution. and for showing computational hardness results we will show the existence of at least one trivial feasible solution.
Number of covered elements:: The number of covered elements and the corresponding selected sets in an optimal solution in Fmc can differ vastly from that in the maximum $k$ -set coverage problem on the same instance. The reason for the discrepancy is because in Fmc one may need to select fewer covered elements to satisfy the coloring constraints.
Exactly $k$ sets vs. at most $k$ sets:: For the maximum $k$ -set coverage problem any solution trivially can use exactly $k$ sets and therefore there is no change to the solution space whether the problem formulation requires exactly $k$ sets or at most $k$ sets. However, the corresponding situation for Fmc is different since it may be non-trivial to convert a feasible solution containing $k^{\prime}<k$ sets to one containing exactly $k$ sets because of the coloring constraints.

Discrepancy minimization problems

Informally, the discrepancy minimization problem for set systems (Min-disc) is orthogonal to unweighted Fmc. Often Min-disc is studied in the context of two colors, say red and blue, and is defined as follows. Like unweighted Fmc we are given $m$ sets over $n$ elements. However, unlike Fmc element colors are not given a priori but the goal to color every element red or blue to minimize the maximum discrepancy over all sets, where the discrepancy of a set is the absolute difference of the number of red and blue elements it contains. The Beck-Fiala theorem BF81 shows that the discrepancy of any set system is at most $2f$ , Spencer showed in S85 that the discrepancy of any set system is $O(\sqrt{n\log(2m/n)}\,)$ , Bansal provided a randomized polynomial time algorithm achieving Spencer’s bound in B10 , and a deterministic algorithm with similar bounds were provided in LRR17 . On the lower bound side, it is possible to construct set systems such that the discrepancy is $\Omega(\sqrt{n}\,)$ C00 . For generalization of the formulation to more than two colors and corresponding results, see for example D02 ; DS99 ; S03 .

Maximization of non-decreasing submodular set functions with linear inequality constraints

Kulik et al. in KST09 provided approximation algorithms for maximizing a non-decreasing submodular set function subject to multiple linear inequality constraints over the elements. Unfortunately, because the linear constraints in Segr-fmc are equality constraints, Segr-fmc cannot be put in the framework of KST09 and the approximation algorithms in KST09 do not directly apply to Segr-fmc.

5 Summary of our contribution

5.1 Feasibility hardness results

Obviously Fmc (resp., Node-fmc) obeys all the inapproximability results for the maximum $k$ -set coverage (resp., $k$ -node coverage) problem. We show in Lemma 1 that determining feasibility of Fmc instances is $\mathsf{NP}$ -complete even under very restricted parameter values; the proofs cover (or can be easily modified to cover) all the spacial cases of Fmc investigated in this paper. However, our subsequent algorithmic results show that even the existence of one feasible solution gives rise to non-trivial approximation bounds for the objective and the coloring constraints.

5.2 Algorithmic results

A summary of our algorithmic results is shown in Table 1. Based on the discussion in the previous section, all of our algorithms assume that at least one feasible solution for the Fmc instance exists.

coloring constraints

approximation

parameter and other restrictions

(if any)

theorem

problem

name

algorithm

name

& type

approximation

ratio

\varepsilon

-approximation

\varepsilon=

randomized

\varepsilon

-approximation

\varepsilon=

strong

randomized

\varepsilon

-approximation

\varepsilon=

{\mathsf{OPT}_{\#}}

\chi

other

Fmc

Alg-large-opt_#

{\mathcal{R}\mathcal{A}\mathfrak{l}\mathfrak{g}}

\varrho(f)

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

3.16f

O(f)

\Omega(\chi\sqrt{n}\log\chi)\,\,\,\,

—

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

Theorem 9.1

Alg-medium-opt_#

{\mathcal{R}\mathcal{A}\mathfrak{l}\mathfrak{g}}

\varrho(f)

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

3.16f

O(f^{2})

\Omega(a\chi\log\chi)

and

O(\chi\sqrt{n\log\chi})

—

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

Theorem 9.1

Alg-small-opt_#

{\mathcal{R}\mathcal{A}\mathfrak{l}\mathfrak{g}}

\varrho(f)

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

3.16f

O(f^{2}\sqrt{a\,\chi{\mathsf{OPT}}_{\#}}\,)

—

O(\max\{1,\,\frac{\log n}{\log m}\})

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

Theorem 9.1

Alg-iter-round

{\mathcal{D}\mathcal{A}\mathfrak{l}\mathfrak{g}}

\nicefrac{{1}}{{f}}

O(f^{2})

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

—

O(1)

at most

k+\frac{\chi-1}{2}

sets

Theorem 10.2

\nicefrac{{1}}{{f}}

O(f^{2}+\chi^{2}f)

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

—

at most

k+\chi-1

sets

Theorem 10.2

Node-fmc

Alg-iter-round

{\mathcal{D}\mathcal{A}\mathfrak{l}\mathfrak{g}}

\nicefrac{{1}}{{2}}

4+4\chi

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

—

O(1)

at most

k+\frac{\chi-1}{2}

sets

Theorem 10.1

\nicefrac{{1}}{{2}}

4+2\chi+4\chi^{2}

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

—

at most

k+\chi-1

sets

Theorem 10.1

Other results: same as Fmc with

f=2

Segr-fmc

Alg-greed-plus

{\mathcal{D}\mathcal{A}\mathfrak{l}\mathfrak{g}}

\varrho

2

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

—

at most

k

sets

Theorem 11.1

\Delta

-bal-fmc

Alg-greedy

{\mathcal{D}\mathcal{A}\mathfrak{l}\mathfrak{g}}

\varrho

O(\Delta f)

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

—

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

Proposition 2

Geom-fmc

Alg-geom

{\mathcal{R}\mathcal{A}\mathfrak{l}\mathfrak{g}}

1-O(\delta)

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

1+\delta

—

d=O(1)

\frac{{\mathsf{OPT}}}{\chi}\geq\eta

Theorem 12.1

{\mathcal{D}\mathcal{A}\mathfrak{l}\mathfrak{g}}

: deterministic algorithm

{\mathcal{R}\mathcal{A}\mathfrak{l}\mathfrak{g}}

: randomized algorithm

\nicefrac{{\mathrm{N}}}{{\mathrm{A}}}

: not applicable —: unrestricted

{\varrho(x)={\big{(}1-\nicefrac{{1}}{{x}}\big{)}}^{x}}>1-\nicefrac{{1}}{{{\mathbf{e}}}}

\varrho=\max\{\varrho(f),\,\varrho(k)\}>1-\nicefrac{{1}}{{{\mathbf{e}}}}

\delta

: any constant in the range (0,1]

\eta

: any constant strictly greater than

1

Table 1: A summary of our algorithmic results. The

O(\cdot)

notation is used when constants are irrelevant or not precisely calculated. Precise definitions of (deterministic)

\varepsilon

-approximation, randomized

\varepsilon

-approximation and strong randomized

\varepsilon

-approximation appear in Section 8.

5.3 Remarks on proof techniques

Distributions on level sets with negative correlations

Our randomized algorithms use the sampling result by Srinivasan S01 which allows one to sample variables satisfying an equality precisely while still ensuring that the variables are negatively correlated and therefore the tail bounds by Panconesi and Srinivasan PS97 can be applied. This allows the randomized algorithms in Theorem 9.1 to select precisely $k$ sets while still preserving the properties of the distribution of variables that are needed for the proof.

Strengthening ${\mathsf{LP}}$ -relaxation via additional inequalities

As we show in Section 9.2, a straightforward ${\mathsf{LP}}$ -relaxation of Fmc based on a corresponding known ${\mathsf{LP}}$ -relaxation of maximum $k$ -set coverage problems does not have an finite integrality gap and therefore unsuitable for further analysis. To get around this, we use an approach similar to what was used by prior researchers (e.g., see the works by Carnes and Shmoys CD08 and Carr et al. CFLP00 ) by introducing extra $O(fn)$ covering inequalities which brings down the integrality gap and allows the results in Theorem 9.1 to go through.

Moreover, we had to separately modify existing constraints of or add new constraints to the basic ${\mathsf{LP}}$ -relaxation for the three algorithms, namely algorithms Alg-small-opt_#, Alg-medium-opt_# and Alg-iter-round. Modifications for Alg-small-opt_# and Alg-medium-opt_# in Theorem 9.1 are done to encode the coloring constraints suitably to optimize their coloring constraint approximation bounds for the corresponding parameter ranges. The modifications for Alg-iter-round in Theorem 10.1 and Theorem 10.2 are necessary for the iterated rounding approach to go through.

Doob martingales and Azuma’s inequality

The analysis of the rounding step of our various ${\mathsf{LP}}$ -relaxations are further complicated by the fact that the random element-selection variables may not be pairwise independent; in fact, it is easy to construct examples in which each element-selection variable may be correlated to about $af$ other element-selection variables, thereby ruling out straightforward use of Chernoff-type tail bounds. For sufficient large ${\mathsf{OPT}}_{\#}$ , we remedy this situation by using Doob martinagales and Azuma’s inequality in the analysis of Alg-large-opt_# in Theorem 9.1.

Iterated rounding of ${\mathsf{LP}}$ -relaxation

The analysis of our deterministic algorithm Alg-iter-round uses the iterated rounding approach originally introduced by Jain in J01 and subsequently used by many researchers (the book by Lau, Ravi and Singh LRS11 provides an excellent overview of the topic). A crucial ingredient of this technique used in our proof is the rank lemma. In order to use this technique, we had to modify the ${\mathsf{LP}}$ -relaxation again. When $\chi=O(1)$ , we can do two exhaustive enumeration steps in polynomial time, giving rise to a somewhat better approximation of the coloring constraints.

Random shifting technique

The analysis of our deterministic algorithm Alg-geom uses the random shifting technique that has been used by prior researchers such as A98 ; M99 .

6 Organization of the paper and proof structures

The rest of the paper is organized as follows.

$\triangleright$

In Section 7 we present our result in Lemma 1 on the computational hardness of finding a feasible solution of Fmc.
$\triangleright$

Based on the results in Section 7, we need to make some minimal assumptions and need to consider appropriate approximate variants of the coloring constraints. They are discussed in Section 8 for the purpose of designing (deterministic or randomized) approximation algorithms.
$\triangleright$
In Section 9 we design and analyze our ${\mathsf{LP}}$ -relaxation based randomized approximation algorithms for Fmc. In particular, in Theorem 9.1 we employ two different ${\mathsf{LP}}$ -relaxation of Fmc and combine three randomized rounding analysis on them to get an approximation algorithm whose approximation qualities depend on the range of relevant parameters.
1. $\triangleright$
  
  Parts of the algorithm and analysis specific to the three algorithms Alg-large-opt_#, Alg-medium-opt_# and Alg-small-opt_# are discussed in Section 9.4, Section 9.5 and Section 9.6, respectively.
2. $\triangleright$
  
  Proposition 1 in Section 9.7 shows that the dependence of the coloring constraint bounds in Theorem 9.1(e)(i)–(ii) on $f$ cannot be completely eliminated by better analysis of our ${\mathsf{LP}}$ -relaxations even for $\chi=2$ .
$\triangleright$
In Section 10 we provide polynomial-time deterministic approximations of Fmc via iterated rounding of a new ${\mathsf{LP}}$ -relaxation. Our approximation qualities depend on the parameters $f$ and $\chi$ .
1. $\triangleright$
  
  For better understanding, we first prove our result for the special case Node-fmc of Fmc in Theorem 10.1 (Section 10.1) and later on describe how to adopt the same approach for Fmc in Theorem 10.2 (Section 10.2).
2. $\triangleright$
  
  The proofs for both Theorem 10.1 and Theorem 10.2 are themselves divided into two parts depending on whether $\chi=O(1)$ or not.
$\triangleright$

In Section 11 we provide deterministic approximation algorithms for two special cases of Fmc, namely Segr-fmc and $\Delta$ -bal-fmc.
$\triangleright$

Section 12 provides the deterministic approximation for Geom-fmc.

Our proofs are structured as follows. A complex proof is divided into subsections corresponding to logical sub-divisions of the proofs and the algorithms therein. Often we provide some informal intuitions behind the proofs (including some intuition about why other approaches may not work, if appropriate) before describing the actual proofs.

7 Computational hardness of finding a feasible solution of Fmc

We show that determining if a given instance of Fmc has even one feasible solution is $\mathsf{NP}$ -complete even in very restricted parameter settings. The relevant parameters of importance for Fmc is $a$ , $f$ and $\chi$ ; Lemma 1 shows that the $\mathsf{NP}$ -completeness result holds even for very small values of these parameters.

Lemma 1

Determining feasibility of an instance of Fmc of $n$ elements is $\mathsf{NP}$ -complete even with the following restrictions:

$\triangleright$

the instances correspond to the unweighted version,
$\triangleright$
the following combinations of maximum set-size $a$ , frequency $f$ and number of colors $\chi$ are satisfied:
1. (a)
  
  $f\in\{1,3\}$ , all but one set contains exactly $3$ elements and all $\chi\geq 2$ ,
2. (b)
  
  the instances correspond to Node-fmc (which implies $f=2$ ), $a=O(\sqrt{n}\,)$ , and all $\chi\geq 2$ , or
3. (c)
  
  $f=1$ , $a=3$ and $\chi=\nicefrac{{n}}{{3}}$ .

Moreover, the following assertions also hold:

$\triangleright$

The instances of Fmc generated in (a) and (b) actually are instances of Segr-fmc.
$\triangleright$

For the instances of Fmc generated in (c), ${\mathsf{OPT}}_{\#}=\chi=\nicefrac{{n}}{{3}}$ and, assuming $P\neq\mathsf{NP}$ , there is no polynomial time approximation algorithm that has either a finite approximation ratio or satisfies the coloring constraints in the $\varepsilon$ -approximate sense $($ cf. eq. (5) $)$ for any finite $\varepsilon$ .

A proof of Lemma 1 appears in the appendix.

Remark 1

It may be tempting to conclude that an approach similar to what is stated below using the $k$ -set coverage problem as a “black box” may make the claims in Lemma 1 completely obvious. We simply take any hard instance of $k$ -set coverage and equi-partition the universe arbitrarily into $\chi$ many color classes and let this be the corresponding instance of Fmc. Using a suitable standard reductions of $\mathsf{NP}$ -hardness the $k$ -set coverage problem, if there is a feasible solution of Fmc then the $k$ sets trivially cover the entire universe and thus trivially satisfy the color constraints but otherwise one may (incorrectly) claim that no $k$ sets cover the universe and so the fairness constraints cannot be satisfied. Additionally, one may be tempted to argue that if one takes a $k$ -set coverage problem instance with any additional structure (e.g., bounded occurrence of universe elements) then the coloring does not affect this additional property at all and hence the property is retained in the $k$ -set coverage problem instance with color constraints.

However, such a generic reduction will fail because it is incorrect and because it will not capture all the special parameter restrictions imposed in Lemma 1. For example:

\triangleright

Even though the $k$ sets may not cover the entire universe, it is still possible that they may satisfy the color constraints. For example, consider the following instance of the $k$ -set coverage problem:

\displaystyle\mathcal{U}=\{u_{1},u_{2},u_{3},u_{4},u_{5},u_{6}\},\,k=2,\,\mathcal{S}_{1}=\{u_{1},u_{2}\},\,\mathcal{S}_{2}=\{u_{3},u_{4}\},\,\mathcal{S}_{3}=\{u_{5},u_{6}\}

Suppose we select the equi-partition $\{u_{1},u_{3},u_{5}\},\{u_{2},u_{4},u_{6}\}$ , thus setting $\mathcal{C}(u_{1})=\mathcal{C}(u_{3})=\mathcal{C}(u_{5})=1$ and $\mathcal{C}(u_{2})=\mathcal{C}(u_{4})=\mathcal{C}(u_{6})=2$ . Then any two selected sets will satisfy the coloring constraints.

$\triangleright$

Consider the requirements of the Fmc instances in part (c). Since $f=1$ , every element occurs in exactly one set and thus the set systems for the $k$ -set coverage problem form a partition of the universe. Such an instance cannot be a hard instance of $k$ -set coverage since it admits a trivial polynomial time solution: sort the sets in non-decreasing order of their cardinalities and simply take the first $k$ sets.

8 Relaxing coloring constraints for algorithmic designs

Based on Lemma 1 we need to make the following minimal assumptions for the purpose of designing approximation algorithms with finite approximation ratios:

(i)

We assume the existence of at least one feasible solution for the given instance of Fmc.
(ii)

We assume that ${\mathsf{OPT}}_{\#}$ is sufficiently large compared to $\chi$ , e.g., ${\mathsf{OPT}}_{\#}\geq c\chi$ for some large constant $c>1$ .

Lemma 1 and the example in Fig. 1 also show that satisfying the color constraint exactly (i.e., requiring $\nicefrac{{p_{i}}}{{p_{j}}}$ to be exactly equal to $1$ for all $i$ and $j$ ) need to be relaxed for the purpose of designing efficient algorithms since non-exact solutions of Fmc may not satisfy these constraints exactly. We define an (deterministic) ${\varepsilon}$ -approximate coloring of Fmc (for some $\varepsilon\geq 1$ ) to be a coloring that satisfies the coloring constraints in the following manner:

\displaystyle\text{\bf Deterministic ${\varepsilon}$-approximate coloring}:\,\,\,\,\forall\,i,j\in\{1,\dots,\chi\}\,:\,p_{i}\leq\varepsilon p_{j}

(5)

Note that (5) automatically implies that $p_{i}\geq p_{j}/\varepsilon$ for all $i$ and $j$ . Thus, in our terminology, a $1$ -approximate coloring corresponds to satisfying the coloring constraints exactly. Finally, if our algorithm is randomized, then the $p_{j}$ ’s could be a random values, and then we will assume that the relevant constraints will be satisfied in expectation or with high probability in an appropriate sense. More precisely, (5) will be modified as follows:⁴⁴4We do not provide a bound on ${\mathbb{E}}[\nicefrac{{p_{i}}}{{p_{j}}}]$ since $\nicefrac{{p_{i}}}{{p_{j}}}=\infty$ when $p_{j}=0$ and $p_{j}$ may be zero with a strictly positive probability, and for arbitrary $\chi$ selecting a set individually for each to avoid this situation in our randomized algorithms may select too many sets.

Randomized ${\varepsilon}$ -approximate coloring:: $\forall\,i,j\in\{1,\dots,\chi\}\,:\,{\mathbb{E}}[p_{i}]\leq\varepsilon\,{\mathbb{E}}[p_{j}]$ (5)^′
Randomized strong ${\varepsilon}$ -approximate coloring:: $\bigwedge_{i,j\in\{1,\dots,\chi\}}\Big{(}\Pr\left[p_{i}\leq\varepsilon p_{j}\right]\Big{)}\geq 1-o(1)$ (5)^′′

Unless otherwise stated explicitly, our algorithms will select exactly $k$ sets.

9 ${\mathsf{LP}}$ -relaxation based randomized approximation algorithms for Fmc

If $k$ is a constant then we can solve Fmc( $\chi,k$ ) exactly in polynomial (i.e., $O(n^{k})$ ) time by exhaustive enumeration, so we assume that $k$ is at least a sufficiently large constant. In this section we will employ two slightly different ${\mathsf{LP}}$ -relaxation of Fmc and combine three randomized rounding analysis on them to get an approximation algorithm whose approximation qualities depend on the range of various relevant parameters. The combined approximation result is stated in Theorem 9.1. In the proof of this theorem no serious attempt was made to optimize most constants since we are mainly interested in the asymptotic nature of the bounds, and to simplify exposition constants have been over-estimated to get nice integers. In the statement of Theorem 9.1 and in its proof we will refer to the three algorithms corresponding to the two ${\mathsf{LP}}$ -relaxations as Alg-small-opt_#, Alg-medium-opt_# and Alg-large-opt_#.

Theorem 9.1

Suppose that the instance of Fmc( $\chi,k$ ) has $n$ elements and $m$ sets. Then, we can design three randomized polynomial-time algorithms Alg-small-opt_#, Alg-medium-opt_# and Alg-large-opt_# with the following properties:

(a)

All the three algorithms select $k$ sets $($ with probability $1)$ .
(b)

All the three algorithms are randomized $\varrho(f)$ -approximation for Fmc, i.e., the expected total weight of the selected elements for both algorithms is at least $\varrho(f)>1-\nicefrac{{1}}{{{\mathbf{e}}}}$ times ${\mathsf{OPT}}$ .
(c)

All the three algorithms satisfy the randomized $\varepsilon$ -approximate coloring constraints $($ cf. Inequality (5) ${}^{\prime})$ for $\varepsilon=O(f)$ , i.e., for all $i,j\in\{1,\dots,\chi\}$ , $\frac{{\mathbb{E}}[p_{i}]}{{\mathbb{E}}[p_{j}]}\leq\frac{2f}{\varrho(f)}<3.16f$ .

(d)

The algorithms satisfy the strong randomized $\varepsilon$ -approximate coloring constraints $($ cf. Equation (5)^′′, i.e., $\bigwedge_{i,j\in\{1,\dots,\chi\}}\big{(}\Pr\left[p_{i}\leq\varepsilon p_{j}\right]\big{)}\geq 1-o(1))$ for the values of $\varepsilon$ , ${\mathsf{OPT}}_{\#}$ and $\chi$ as shown below:

	$\varepsilon$	range of ${\mathsf{OPT}}_{\#}$	range of $\chi$	algorithm
(i)	$O(f)$	$\Omega(\chi\sqrt{n}\log\chi)$	unrestricted	Alg-large-opt_#
(ii)	$O(f^{2})$	$\Omega(a\chi\log\chi)$ and $O(\chi\sqrt{n\log\chi})$	unrestricted	Alg-medium-opt_#
(iii)	$O(f^{2}\sqrt{a\,\chi{\mathsf{OPT}}_{\#}}\,)$	unrestricted	$\chi=O(\max\{1,\,\frac{\log n}{\log m}\})$	Alg-small-opt_#

Remark 2

Note that the high-probability $\varepsilon=O(f)$ bound in Theorem 9.1(d)(i) is asymptotically the same as the “ratio of expectation” bound in Theorem 9.1(c).

Remark 3

The dependence on $f$ of the bounds for $\varepsilon$ in Theorem 9.1(d)(i)–(ii) can be contrasted with the Beck-Fiala theorem in discrepancy minimization that shows that the discrepancy of any set system is at most $2f$ .

Remark 4

Consider the special case Node-fmc with $\chi=O(1)$ : for this case $f=2$ and $a$ is equal to the maximum node-degree $\mathsf{deg}_{\max}$ in the graph. The bounds in Theorem 9.1(d)(i)–(ii) for this special case imply a $O(1)$ -approximation of color constraints unless ${\mathsf{OPT}}_{\#}$ is not sufficiently large compared to $\mathsf{deg}_{\max}$ . To illustrate the bound for smaller ${\mathsf{OPT}}_{\#}$ in Theorem 9.1(d)(iii), if ${\mathsf{OPT}}_{\#}=\mathsf{deg}_{\max}^{({1}/{2})-\varepsilon}$ for some $\varepsilon>0$ then the approximation bound of the coloring constraints is $O(\mathsf{deg}_{\max}^{1-\varepsilon})$ .

A proof of Theorem 9.1 is discussed in the remaining subsections of this section. The following notations will be used uniformly throughout the proof.

$\triangleright$

$x_{1},\dots,x_{n}\in\{0,1\}$ and $y_{1},\dots,y_{m}\in\{0,1\}$ are the usual indicator variables for the elements $u_{1},\dots,u_{n}$ and the sets $\mathcal{S}_{1},\dots,\mathcal{S}_{m}$ , respectively, Their values in an optimal solution of the ${\mathsf{LP}}$ -relaxation under consideration will be denoted by $x_{1}^{\ast},\dots,x_{n}^{\ast}$ and $y_{1}^{\ast},\dots,y_{m}^{\ast}$ , respectively.
$\triangleright$

$C_{j}$ is the set of all elements colored $j$ for $j\in\{1,\dotsc,\chi\}$ in the given instance of Fmc.
$\triangleright$

${\mathsf{OPT}}_{\mathrm{frac}}$ is the optimum value of the objective function of the ${\mathsf{LP}}$ -relaxation under consideration.

9.1 An obvious generalization of ${\mathsf{LP}}$ -relaxation of maximum $k$ -set coverage fails

maximize	$\sum_{i=1}^{n}w(u_{i})x_{i}$
subject to	$x_{j}\leq\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}$	for $j=1,\dots,n$
	$\sum_{\ell=1}^{m}y_{\ell}=k$
	$0\leq x_{j}\leq 1$	for $j=1,\dots,n$
	$0\leq y_{\ell}\leq 1$	for $\ell=1,\dots,m$

Figure 2: A well-known

{\mathsf{LP}}

-relaxation of the element-weighted maximum

k

-set coverage problem.

It is well-known that the ${\mathsf{LP}}$ -relaxation for the element-weighted maximum $k$ -set coverage problem as shown in Fig. 2 followed by a suitable deterministic or randomized rounding provides an optimal approximation algorithm for the problem (e.g., see AS04 ; MR95 ). A straightforward way to extend this ${\mathsf{LP}}$ -relaxation is to add the following ${\chi(\chi-1)}/{2}$ additional constraints, one corresponding to each pair of colors:

\sum_{u_{\ell}\in C_{i}}x_{\ell}=\!\!\sum_{u_{\ell}\in C_{j}}x_{\ell}\text{ for }i,j\in\{1,\dots,\chi\},i<j

Unfortunately, this may not lead to a $\varepsilon$ -approximate coloring (cf. Equation (5)) for any non-trivial $\varepsilon$ as the following example shows. Suppose our instance of an unweighted Fmc( $2,2$ ) has four sets $\mathcal{S}_{1}=\{u_{1}\}$ , $\mathcal{S}_{2}=\{u_{2},\dots,u_{n-2}\}$ , $\mathcal{S}_{3}=\{u_{n-1}\}$ and $\mathcal{S}_{4}=\{u_{n}\}$ with the elements $u_{1}$ and $u_{n-1}$ having color $1$ and all other elements having color $2$ . Clearly the solution to this instance consists of the sets $\mathcal{S}_{3}$ and $\mathcal{S}_{4}$ with ${\mathsf{OPT}}=2$ . On the other hand, the fractional solution $y_{1}^{\ast}=y_{2}^{\ast}=x_{1}^{\ast}=x_{2}^{\ast}=1$ and all remaining variables being zero is also an optimal solution of the ${\mathsf{LP}}$ -relaxation, but any rounding approach that does not change the values of zero-valued variables in the fractional solution must necessarily result in an integral solutions in which $p_{2}/p_{1}=n-3$ . The example is easily generalized for arbitrary $k$ .

9.2 Alg-large-opt_#: strengthening ${\mathsf{LP}}$ -relaxation via additional inequalities

One problem that the ${\mathsf{LP}}$ -relaxation in Fig. 2 faces when applied to Fmc is the following. We would like each element-indicator variable $x_{j}$ to satisfy $x_{j}^{\ast}=\min\big{\{}1,\,\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}^{\ast}\big{\}}$ in an optimum solution of the ${\mathsf{LP}}$ , but this may not be true as shown in the simple example in the previous section. Past researchers have corrected this kind of situation by introducing extra valid inequalities that hold for any solution to the problem but restrict the feasible region of the ${\mathsf{LP}}$ . For example, Carnes and Shmoys in CD08 and Carr et al. in CFLP00 introduced a set of additional inequalities, which they called the KC (Knapsack Cover) inequalities, to strengthen the integrality gaps of certain types of capacitated covering problems. Following their ideas, we add the extra $O(fn)$ “covering inequalities” which are satisfied by any integral solution of the ${\mathsf{LP}}$ :

\displaystyle x_{j}\geq y_{\ell}\text{ for }j=1,\dots,n,\,\ell=1,\dots,m,\text{ and }u_{j}\in\mathcal{S}_{\ell}

In addition, we adjust our ${\mathsf{LP}}$ -relaxation in the following manner. Since ${\mathsf{OPT}}_{\#}$ is an integer from $\{\chi,2\chi,\dots,(\lfloor\nicefrac{{n}}{{\chi}}\rfloor)\chi\}$ , we can “guess” the correct value of ${\mathsf{OPT}}_{\#}$ by running the algorithm for each of the $\lfloor\nicefrac{{n}}{{\chi}}\rfloor$ possible value of ${\mathsf{OPT}}_{\#}$ , consider those solutions that maximized its objective function and select that one among these solutions that has the largest value of ${\mathsf{OPT}}_{\#}$ . Thus, we may assume that our ${\mathsf{LP}}$ -relaxation knows the value of ${\mathsf{OPT}}_{\#}$ exactly, and we add the following extra equality:

$\sum_{i=1}^{n}x_{i}={\mathsf{OPT}}_{\#}$

The resulting ${\mathsf{LP}}$ -relaxation is shown in its completeness in Fig. 3 for convenience.

maximize	$\sum_{i=1}^{n}w(u_{i})x_{i}$
subject to	$x_{j}\leq\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}$	for $j=1,\dots,n$
	$\sum_{\ell=1}^{m}y_{\ell}=k$
	$x_{j}\geq y_{\ell}$	for $j=1,\dots,n$ , $\ell=1,\dots,m$ , and $u_{j}\in\mathcal{S}_{\ell}$
	$\sum_{i=1}^{n}x_{i}={\mathsf{OPT}}_{\#}$
	$\sum_{u_{\ell}\in C_{i}}x_{\ell}=\sum_{u_{\ell}\in C_{j}}x_{\ell}$	for $i,j\in\{1,\dots,\chi\}$ , $i<j$
	$0\leq x_{j}\leq 1$	for $j=1,\dots,n$
	$0\leq y_{\ell}\leq 1$	for $\ell=1,\dots,m$

Figure 3: A

{\mathsf{LP}}

-relaxation for Alg-large-opt_# with

n+m

variables and

O\left(fn+\chi^{2}\right)

constraints.

9.3 A general technique for obtaining joint high-probability statement

Suppose that our randomized ${\mathsf{LP}}$ -relaxation based algorithm guarantees that $\Pr\left[p_{i}/p_{j}>\varepsilon\right]<\nicefrac{{1}}{{(c\chi^{2})}}$ for some constant $c\geq 3$ independently for all $i,j\in\{1,\dots,\chi\}$ . Then

\bigwedge_{i,j\in\{1,\dots,\chi\}}\Pr\left[p_{i}\leq\varepsilon\,p_{j}\right]=1-\bigvee_{i,j\in\{1,\dots,\chi\}}\Pr\left[p_{i}>\varepsilon\,p_{j}\right]\\ \geq 1-\sum_{i,j\in\{1,\dots,\chi\}}\Pr\left[p_{i}\geq\varepsilon\,p_{j}\right]>1-(\chi^{2})\frac{1}{c\chi^{2}}=1-\frac{1}{c}=c^{\prime}

To boost the success probability, we repeat the randomized rounding $c^{\prime}\ln n$ times, compute the quantity $\sigma=\max_{i,j\in\{1,\dots,\chi\},\,p_{j}\neq 0}\{\nicefrac{{p_{i}}}{{p_{j}}}\}$ in each iteration, and output the solution in that iteration that resulted in the minimum value of $\sigma$ . It then follows that for the selected solution satisfies the strong randomized $\varepsilon$ -approximate coloring constraints since $\bigwedge_{i,j\in\{1,\dots,\chi\}}\Pr\left[p_{i}\leq\varepsilon\,p_{j}\right]\geq 1-\left(\nicefrac{{1}}{{c^{\prime}}}\right)^{c^{\prime}\ln n}>1-\nicefrac{{1}}{{n^{2}}}$ .

9.4 Alg-large-opt_#: further details and relevant analysis

For our randomized rounding approach, we recall the following result from S01 .

Fact 1

S01 Given numbers $p_{1},\dots,p_{r}\in[0,1]$ such that $\ell=\sum_{i=1}^{r}p_{i}$ is an integer, there exists a polynomial-time algorithm that generates a sequence of integers $X_{1},\dots,X_{r}$ such that $(a)$ $\sum_{i=1}^{r}X_{i}$ with probability $1$ , $(b)$ $\Pr\left[X_{i}=1\right]=p_{i}$ for all $i\in\{1,\dots,r\}$ , and $(c)$ for any real numbers $\alpha_{1},\dots,\alpha_{r}\in[0,1]$ the sum $\sum_{i=1}^{r}\alpha_{i}X_{i}$ satisfies standard Chernoff bounds.

We round $y_{1}^{\ast},\dots,y_{m}^{\ast}$ to $y_{1}^{+},\dots,y_{m}^{+}$ using the algorithm mentioned in Fact 1; this ensures $\sum_{\ell=1}^{m}y_{\ell}^{+}=\sum_{\ell=1}^{m}y_{\ell}^{\ast}=k$ resulting in selection of exactly $k$ sets. This proves the claim in (b). We round $x_{1}^{\ast},\dots,x_{n}^{\ast}$ to $x_{1}^{+},\dots,x_{n}^{+}$ in the following way: for $j=1,\dots,n$ , if $u_{j}\in\mathcal{S}_{\ell}$ for some $y_{\ell}^{+}=1$ then set $x_{j}^{+}=1$ .

Proof of (b)

Our proof of (c) is similar to that for the maximum $k$ -set coverage and is included for the sake of completeness. Note that $x_{j}^{+}=0$ if and only if $y_{\ell}^{+}=0$ for every set $\mathcal{S}_{\ell}$ containing $u_{j}$ and thus:

\textstyle{\mathbb{E}}[x_{j}^{+}]=\Pr[x_{j}^{+}=1]=1-\Pr[x_{j}^{+}=0]=1-\prod_{u_{j}\in\mathcal{S}_{\ell}}\Pr[y_{\ell}^{+}=0]=1-\prod_{u_{j}\in\mathcal{S}_{\ell}}(1-y_{\ell}^{\ast})\\ \textstyle\geq 1-\Big{(}\frac{\sum_{u_{j}\in\mathcal{S}_{\ell}}\big{(}1-y_{\ell}^{\ast}\big{)}}{f_{j}}\Big{)}^{f_{j}}=1-\Big{(}1-\frac{\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}^{\ast}}{f_{j}}\Big{)}^{f_{j}}\geq 1-\Big{(}1-\frac{\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}^{\ast}}{f}\Big{)}^{f}

(6)

where we have used inequality (3). If $\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}^{\ast}\geq 1$ then obviously (6) implies ${\mathbb{E}}[x_{j}^{+}]\geq\varrho(f)\geq\varrho(f)x_{j}^{\ast}$ . Otherwise, $x_{j}^{\ast}\leq\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}^{\ast}<1$ and then by (4) we get

\displaystyle\textstyle{\mathbb{E}}[x_{j}^{+}]\geq 1-\left(1-\frac{\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}^{\ast}}{f}\right)^{f}>1-\left(1-\frac{x_{j}^{\ast}}{f}\right)^{f}\geq\varrho(f)x_{j}^{\ast}

(7)

This implies our bound since

\displaystyle\textstyle{\mathbb{E}}[\sum_{i=1}^{n}w(u_{i})x_{i}^{+}]=\sum_{i=1}^{n}w(u_{i}){\mathbb{E}}[x_{i}^{+}]\geq\sum_{i=1}^{n}w(u_{i})\varrho(f)x_{j}^{\ast}=\varrho(f){\mathsf{OPT}}_{\mathrm{frac}}\geq\varrho(f){\mathsf{OPT}}

Proof of (c)

Note that inequalities (1) and (2) imply $1-x\leq{\mathbf{e}}^{-x}\leq 1-x+(x^{2}/2)$ for all $x\in[0,1]$ . In particular, the following implication holds:

\displaystyle\forall\,c>1\,\,\forall\,x\in[0,(2/c^{2})(c-1)]\,:\,1-x\geq 1-cx+(c^{2}x^{2}/2)\geq{\mathbf{e}}^{-cx}

(8)

We estimate an upper bound on ${\mathbb{E}}[x_{j}^{+}]$ in terms of $x_{j}^{\ast}$ in the following manner:

Case 1: $\boldsymbol{\exists\,\ell}$ such that $\boldsymbol{u_{j}\in\mathcal{S}_{\ell}}$ and $\boldsymbol{y_{\ell}^{\ast}>\nicefrac{{1}}{{2}}}$ .

Thus, $x_{j}^{\ast}\geq y_{\ell}^{\ast}>\nicefrac{{1}}{{2}}$ , and ${\mathbb{E}}[x_{j}^{+}]\leq 1\leq 2x_{j}^{\ast}$ .

Case 2: $\boldsymbol{y_{\ell}^{\ast}\leq\nicefrac{{1}}{{2}}}$ for every index $\boldsymbol{\ell}$ satisfying $\boldsymbol{u_{j}\in\mathcal{S}_{\ell}}$ .

Note that $x_{j}^{\ast}\geq\left(\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}^{\ast}\right)/f$ , and setting $c=2$ in inequality (8) we get $1-x\geq{\mathbf{e}}^{-2x}$ for all $x\in[0,\nicefrac{{1}}{{2}}]$ . Now, standard calculations show the following:

{\mathbb{E}}[x_{j}^{+}]=1-\prod_{u_{j}\in\mathcal{S}_{\ell}}(1-y_{\ell}^{\ast})\leq 1-\prod_{u_{j}\in\mathcal{S}_{\ell}}{\mathbf{e}}^{-2y_{\ell}^{\ast}}=1-{\mathbf{e}}^{-2\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}^{\ast}}\\ =1-{\mathbf{e}}^{-2fx_{j}^{\ast}}\leq 1-(1-2fx_{j}^{\ast})=2fx_{j}^{\ast}

Combining all the cases and using (7), it follows that $\varrho(f)x_{j}^{\ast}\leq{\mathbb{E}}[x_{j}^{+}]\leq\min\big{\{}1,\,2fx_{j}^{\ast}\big{\}}$ . Recall that $\sum_{u_{\ell}\in C_{j}}x_{\ell}^{\ast}=\nicefrac{{{\mathsf{OPT}}_{\#}}}{{\chi}}>0$ for every $j\in\{1,\dots,\chi\}$ . Since ${\mathbb{E}}[p_{j}]={\mathbb{E}}[\sum_{u_{\ell}\in C_{j}}x_{\ell}^{+}]=\sum_{u_{\ell}\in C_{j}}{\mathbb{E}}[x_{\ell}^{+}]$ , we get the following bounds for all $j\in\{1,\dots,\chi\}$ :

\displaystyle\textstyle\varrho(f)\frac{{\mathsf{OPT}}_{\#}}{\chi}=\sum_{u_{\ell}\in C_{j}}\varrho(f)x_{\ell}^{\ast}\leq{\mathbb{E}}[p_{j}]=\sum_{u_{\ell}\in C_{j}}{\mathbb{E}}[x_{\ell}^{+}]\leq\sum_{u_{\ell}\in C_{j}}(2f)x_{\ell}^{\ast}=2f\frac{{\mathsf{OPT}}_{\#}}{\chi}

(9)

which gives the bound $\frac{{\mathbb{E}}[c_{i}]}{{\mathbb{E}}[c_{j}]}\leq\frac{2f}{\varrho(f)}$ for all $i,j\in\{1,\dots,\chi\}$ .

Proofs of (d)(i) via Doob martingales

Note that the random variables $x_{1}^{+},\dots,x_{n}^{+}$ may not be pairwise independent since two distinct elements belonging to the same set are correlated, and consequently the random variables $p_{1},\dots,p_{\chi}$ also may not be pairwise independent. Indeed in the worst case an element-selection variable may be correlated to $(a-1)f$ other element-selection variables, thereby ruling out straightforward use of Chernoff-type tail bounds.

For sufficient large ${\mathsf{OPT}}_{\#}$ , this situation can be somewhat remedied by using Doob martinagales and Azuma’s inequality by finding a suitable ordering of the element-selection variables conditional on the rounding of the set-selection variables. We assume that the reader is familiar with basic definitions and results for the theory of martingales (e.g., see (MR95, , Section $4.4$ )). Fix an arbitrary ordered sequence $y_{1}^{+},\dots,y_{m}^{+}$ of the set-indicator variables. Call an element-indicator variable $x_{i}^{+}$ “settled” at the $t^{\mathrm{th}}$ step if and only if $\cup_{u_{i}\in\mathcal{S}_{j}}\{y_{j}\}\not\subseteq\{y_{1}^{+},\dots,y_{t-1}^{+}\}$ and $\cup_{u_{i}\in\mathcal{S}_{j}}\{y_{j}\}\subseteq\{y_{1}^{+},\dots,y_{t}^{+}\}$ . The elementary event in our underlying sample space $\boldsymbol{\Omega}$ are all possible $2^{n}$ assignments of $0$ - $1$ values to the variables $x_{1}^{+},\dots,x_{n}^{+}$ . For each $t\in\{1,\dots,m\}$ , let $V_{t}$ be the subset of element-selection variables whose values are settled at the $t^{\mathrm{th}}$ step, let $\pi_{t}$ be an arbitrary ordering of the variables in $V_{t}$ , and let us relabel the element-indicator variable names so that $x_{1}^{+},x_{2}^{+},\dots,x_{n}^{+}$ be the ordering of all element-selection variables given by the ordering $\pi_{1},\dots,\pi_{m}$ . For each $t\in\{0,1,\dots,m\}$ and each $w_{1},\dots,w_{t}\in\{0,1\}$ , let $\mathbf{B}_{w_{1},\dots,w_{t}}$ denote the event that $y_{j}^{+}=w_{j}$ for $j\in\{1,\dots,t\}$ . Let $x_{1}^{+},x_{2}^{+},\dots,x_{q_{t}}^{+}$ be the union of set of all $q_{t}$ element-indicator variables that are settled at the $i^{\mathrm{th}}$ step over all $i\in\{1,\dots,t\}$ , and suppose that the event $\mathbf{B}_{w_{1},\dots,w_{t}}$ induces the following assignment of values to the element-indicator variables: $x_{1}^{+}=b_{1},\dots,x_{q_{t}}^{+}=b_{q_{t}}$ for some $b_{1},\dots,b_{q_{t}}\in\{0,1\}$ . Define the block $\mathbf{B}^{\prime}_{w_{1},\dots,w_{t}}\subseteq\boldsymbol{\Omega}$ induced by the event $\mathbf{B}_{w_{1},\dots,w_{t}}$ as $\mathbf{B}^{\prime}_{w_{1},\dots,w_{t}}=\{b_{1},\dots,b_{q_{t}}r_{q_{t}+1}\dots r_{n}\,|\,r_{q_{t}+1},\dots,r_{n}\in\{0,1\}\}$ . Letting $\boldsymbol{\mathbb{F}}_{t}$ be the $\sigma$ -field generated by the partition of $\boldsymbol{\Omega}$ into the blocks $\mathbf{B}^{\prime}_{w_{1},\dots,w_{t}}$ for each $w_{1},\dots,w_{t}\in\{0,1\}$ , it follows that $\boldsymbol{\mathbb{F}}_{0},\boldsymbol{\mathbb{F}}_{1},\dots,\boldsymbol{\mathbb{F}}_{m}$ form a filter for the $\sigma$ -field $(\boldsymbol{\Omega},2^{\boldsymbol{\Omega}})$ . Suppose that $\mathcal{U}$ contains $n_{i}>0$ elements of color $i$ , let $x_{\alpha_{1}}^{+},\dots,x_{\alpha_{n_{i}}}^{+}$ be the ordered sequence of the element-selection variables for elements of color $i$ determined by the subsequence of these variables in the ordering $x_{1}^{+},\dots,x_{n}^{+}$ , and suppose that $x_{\alpha_{j}}^{+}$ was settled at the $t_{\alpha_{j}}^{\mathrm{th}}$ step. Let $X^{i}=\sum_{j=1}^{n_{i}}x_{\alpha_{j}}^{+}$ , and define the Doob martingale sequence $X_{0},X_{1},\dots,X_{n_{i}}$ where $X_{0}={\mathbb{E}}[X^{i}]=\sum_{j=1}^{n_{i}}{\mathbb{E}}[x_{\alpha_{j}}^{+}]$ , and $X_{\ell}={\mathbb{E}}[X^{i}\,|\,y_{1}^{+},\dots,y_{t_{\alpha_{\ell}}}^{+}]$ for all $\ell\in\{1,\dots,n_{i}\}$ . Since $n_{i}<n$ , $X_{n_{i}}=X^{i}$ and $|X_{\ell}-X_{\ell-1}|\leq 1$ for for all $\ell\in\{1,\dots,n_{i}\}$ , by Azuma’s inequality (for any $\Delta>{\mathbf{e}}$ ),

\Pr\left[\,\left|{\textstyle\sum_{j=1}^{n_{i}}}x_{\alpha_{j}}^{+}-{\textstyle\sum_{j=1}^{n_{i}}}{\mathbb{E}}[x_{\alpha_{j}}^{+}]\right|\geq 3\sqrt{\ln\Delta}\sqrt{n}\right]\leq\Pr\left[\,\left|{\textstyle\sum_{j=1}^{n_{i}}}x_{\alpha_{j}}^{+}-X_{0}\right|\geq 3\sqrt{\ln\Delta}\sqrt{n_{i}}\right]\\ =\Pr\left[|X_{n_{i}}-X_{0}|\geq 3\sqrt{\ln\Delta}\sqrt{n_{i}}\right]\leq{\mathbf{e}}^{-4\ln\Delta}=\Delta^{-4}\\ \Rightarrow\,\Pr\left[\,{\textstyle\sum_{j=1}^{n_{i}}}{\mathbb{E}}[x_{\alpha_{j}}^{+}]-3\sqrt{\ln\Delta}\sqrt{n}<{\textstyle\sum_{j=1}^{n_{i}}}x_{\alpha_{j}}^{+}<{\textstyle\sum_{j=1}^{n_{i}}}{\mathbb{E}}[x_{\alpha_{j}}^{+}]+3\sqrt{\ln\Delta}\sqrt{n}\right]>1-\Delta^{-4}

Recall from (9) that $(1-\nicefrac{{1}}{{{\mathbf{e}}}})\frac{{\mathsf{OPT}}_{\#}}{\chi}<\sum_{j=1}^{n_{i}}{\mathbb{E}}[x_{\alpha_{j}}^{+}]\leq 2f\frac{{\mathsf{OPT}}_{\#}}{\chi}$ , and the inequality $\sum_{j=1}^{n_{i}}{\mathbb{E}}[x_{\alpha_{j}}^{+}]\geq 6\sqrt{\ln\Delta}\sqrt{n}$ is therefore satisfied provided ${\mathsf{OPT}}\geq 6\big{(}1-\nicefrac{{1}}{{{\mathbf{e}}}}\big{)}^{-1}(\sqrt{n\ln\Delta})\chi$ . Setting $\Delta=2\chi$ and remembering that $p_{i}={\textstyle\sum_{j=1}^{n_{i}}}x_{\alpha_{j}}^{+}$ , we get the following bound for any $i\in\{1,\dots,\chi\}$ :

\displaystyle\Pr\left[\,\textstyle{\frac{1-\nicefrac{{1}}{{{\mathbf{e}}}}}{2}}\left(\frac{{\mathsf{OPT}}_{\#}}{\chi}\right)<p_{i}<3f\left(\frac{{\mathsf{OPT}}_{\#}}{\chi}\right)\right]>1-(1/16)\chi^{-4}

and therefore $\Pr\left[\frac{p_{i}}{p_{j}}\leq\frac{6}{1-\nicefrac{{1}}{{{\mathbf{e}}}}}f\right]>1-(1/8)\chi^{-4}$ . Thus, it follows that

\bigwedge_{i,j\in\{1,\dots,\chi\}}\Pr\left[p_{i}\leq{\textstyle\frac{6}{1-\nicefrac{{1}}{{{\mathbf{e}}}}}}\,p_{j}\right]=1-\bigvee_{i,j\in\{1,\dots,\chi\}}\Pr\left[p_{i}>{\textstyle\frac{6}{1-\nicefrac{{1}}{{{\mathbf{e}}}}}}\,p_{j}\right]\\ \geq 1-\sum_{i,j\in\{1,\dots,\chi\}}\Pr\left[p_{i}\geq{\textstyle\frac{6}{1-\nicefrac{{1}}{{{\mathbf{e}}}}}}\,p_{j}\right]>{\textstyle 1-(\chi^{2})\frac{1}{8\chi^{4}}\geq\frac{1}{32}}

This implies our claim in (d)(i) using the technique in Section 9.3.

9.5 Alg-medium-opt_#: details and proofs of relevant claims in (a)–(c) and (d)(ii)

Alg-medium-opt_#: idea behind the modified ${\mathsf{LP}}$ -relaxation and approach

A limitation of Alg-large-opt_# is that we could not use Fact 1 of Srinivasan to the fullest extent. Although Fact 1 guaranteed that the set-indicator variables are negatively correlated and hence Chernoff-type tail bounds can be applied to them due to the result by Panconesi and Srinivasan PS97 , our coloring constraints are primarily indicated by element-indicator variables which depend implicitly on the set-indicator variables. In fact, it is not difficult to see that the element-indicator variables are not negatively correlated in the sense of PS97 ; S01 ⁵⁵5A set of binary random variables $z_{1},\dots,z_{r}\in\{0,1\}$ are called negatively correlated in PS97 ; S01 if the following holds: $\forall\,I\subseteq\{1,\dots,r\}:\,\Pr\left[\wedge_{i\in I}(z_{i}=0)\right]\leq\prod_{i\in I}\Pr\left[z_{i}=0\right]\text{ and }\Pr\left[\wedge_{i\in I}(z_{i}=1)\right]\leq\prod_{i\in I}\Pr\left[z_{i}=1\right]$ even if the set-indicator variables are negatively correlated.

Our idea is to remedy the situation by expressing the coloring constraints also by set-indicator variables and use the element-indicator variables to implicitly control the set-indicator variables in these coloring constraints. This will also necessitate using additional variables.

A modification of the ${\mathsf{LP}}$ -relaxation in Fig. 3

To begin, we quantify the number of elements of different colors in a set $\mathcal{S}_{i}$ using the following notation: for $j\in\{1,\dots,\chi\}$ , let $\nu_{i,j}$ be the number of elements in $\mathcal{S}_{i}$ of color $j$ . Note that $0\leq\nu_{i,j}\leq a$ . Fix an optimal integral solution of Fmc( $\chi,k$ ) covering ${\mathsf{OPT}}_{\#}$ elements and a color value $j$ , and consider the following two quantities: $\mathcal{A}=\sum_{u_{\ell}\in C_{j}}x_{\ell}$ and $\mathcal{B}=\sum_{i=1}^{m}\nu_{i,j}\,y_{i}$ . Note that $\mathcal{A}=\frac{{\mathsf{OPT}}_{\#}}{\chi}$ , $\mathcal{B}\in\{k,k+1,\dots,ak\}$ , and $\mathcal{A}\leq\mathcal{B}\leq f\mathcal{A}$ by definition of $f$ . Thus, $\mathcal{B}={\mathfrak{h}}_{j}{\mathsf{OPT}}_{\#}$ is satisfied by a ${\mathfrak{h}}_{j}$ that is a rational number from the set $\big{\{}\frac{1}{\chi},\frac{1}{\chi}+\frac{1}{{\mathsf{OPT}}_{\#}},\frac{1}{\chi}+\frac{2}{{\mathsf{OPT}}_{\#}},\dots,\frac{f}{\chi}-\frac{1}{{\mathsf{OPT}}_{\#}},\frac{f}{\chi}\big{\}}$ . We will use the ${\mathsf{LP}}$ -relaxation in Fig. 3 with $\chi$ additional variables ${\mathfrak{h}}_{1},\dots,{\mathfrak{h}}_{\chi}$ and the following additional constraints

$\sum_{i=1}^{m}\nu_{i,j}\,y_{i}={\mathfrak{h}}_{j}{\mathsf{OPT}}_{\#}$	for $j\in\{1,\dots,\chi\}$
$\nicefrac{{1}}{{\chi}}\leq{\mathfrak{h}}_{j}\leq\nicefrac{{f}}{{\chi}}$	for $j=1,\dots,\chi$

For reader’s convenience, the new ${\mathsf{LP}}$ -relaxation in its entirety is shown in Fig. 4.

maximize	$\sum_{i=1}^{n}w(u_{i})x_{i}$
subject to	$x_{j}\leq\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}$	for $j=1,\dots,n$
	$\sum_{\ell=1}^{m}y_{\ell}=k$
	$x_{j}\geq y_{\ell}$	for $j=1,\dots,n$ , $\ell=1,\dots,m$ , and $u_{j}\in\mathcal{S}_{\ell}$
	$\sum_{i=1}^{n}x_{i}={\mathsf{OPT}}_{\#}$
	$\sum_{i=1}^{m}\nu_{i,j}\,y_{i}={\mathfrak{h}}_{j}{\mathsf{OPT}}_{\#}$	for $j\in\{1,\dots,\chi\}$
	$\sum_{u_{\ell}\in C_{i}}x_{\ell}=\sum_{u_{\ell}\in C_{j}}x_{\ell}$	for $i,j\in\{1,\dots,\chi\}$ , $i<j$
	$0\leq x_{j}\leq 1$	for $j=1,\dots,n$
	$0\leq y_{\ell}\leq 1$	for $\ell=1,\dots,m$
	$\nicefrac{{1}}{{\chi}}\leq{\mathfrak{h}}_{j}\leq\nicefrac{{f}}{{\chi}}$	for $j=1,\dots,\chi$

Figure 4: A modified

{\mathsf{LP}}

-relaxation for Alg-medium-opt_# with with

n+m+\chi

variables and

O\left(fn+\chi^{2}\right)

constraints.

Analysis of the modified ${\mathsf{LP}}$ -relaxation

By our assumption on ${\mathfrak{h}}_{j}$ ’s, the ${\mathsf{LP}}$ -relaxation has a feasible solution. We use the same randomized rounding procedure (using Fact 1) as in Section 9.4 for Alg-large-opt_#. The proofs for parts (b)–(d) are the same as before since all prior relevant inequalities are still included. Thus, we concentrate on the proof of (e)(ii). A crucial thing to note is the following simple observation:

Consider the sum $\Delta=\sum_{i=1}^{m}\nu_{i,j}\,y_{i}$ for any assignment of values $y_{1},\dots,y_{m}\in\{0,1\}$ . Then, the number of elements covered by the sets corresponding to those variables that are set to $1$ is between $\Delta/f$ and $\Delta$ .

Fix a color $j$ . Let $\mathcal{K}_{j}={\mathfrak{h}}_{j}{\mathsf{OPT}}_{\#}$ , $\alpha_{i}=\frac{\nu_{i,j}}{a}$ and consider the summation $\mathcal{L}_{j}=\sum_{i=1}^{m}\alpha_{i}\,y_{i}^{+}$ . Since $\alpha_{i}\in[0,1]$ for all $i$ , by Fact 1 we can apply standard Chernoff bounds MR95 for $\mathcal{L}_{j}$ . Note that ${\mathbb{E}}[\mathcal{L}_{j}]=\sum_{i=1}^{m}\alpha_{i}\,y_{i}^{\ast}=\frac{\mathcal{K}_{j}}{a}$ . Assuming $\mathcal{K}_{j}\geq 16a\ln\chi$ , we get the following for the tail-bounds:

	$\displaystyle\Pr\left[{\textstyle\sum_{i=1}^{m}\nu_{i,j}\,y_{i}^{+}>5\mathcal{K}_{j}}\right]=\Pr\left[\mathcal{L}_{j}>5(\nicefrac{{\mathcal{K}_{j}}}{{a}})\right]<2^{-{6\mathcal{K}_{j}}/{a}}\leq 2^{-96\ln\chi}<{\chi}^{-96}$
	$\displaystyle\Pr\left[{\textstyle\sum_{i=1}^{m}\nu_{i,j}\,y_{i}^{+}<\nicefrac{{\mathcal{K}_{j}}}{{2}}}\right]=\Pr\left[{\textstyle\mathcal{L}_{j}<\frac{{\mathcal{K}_{j}}/{a}}{2}}\right]<{\mathbf{e}}^{-\frac{\mathcal{K}_{j}}{8a}}\leq{\mathbf{e}}^{-2\ln\chi}={\chi}^{-2}$

Remember that $p_{j}=\sum_{u_{\ell}\in C_{j}}x_{\ell}^{+}$ is the random variable denoting the number of elements of color $j$ selected by our randomized algorithm. Since $\frac{1}{f}\sum_{i=1}^{m}\nu_{i,j}\,y_{i}^{+}\leq c_{j}\leq\sum_{i=1}^{m}\nu_{i,j}\,y_{i}^{+}$ we get

	$\displaystyle\Pr\left[p_{j}>5\mathcal{K}_{j}\right]\leq\Pr\left[\textstyle\sum_{i=1}^{m}\nu_{i,j}\,y_{i}^{+}>5\mathcal{K}_{j}\right]<{\chi}^{-96},$
	$\displaystyle\Pr\left[\textstyle p_{j}<\frac{\mathcal{K}_{j}}{2f}\right]\leq\Pr\left[\textstyle\sum_{i=1}^{m}\nu_{i,j}\,y_{i}^{+}<\frac{\mathcal{K}_{j}}{2}\right]<{\chi}^{-2}$

Note that $\frac{{\mathsf{OPT}}_{\#}}{\chi}\leq\mathcal{K}_{j}={\mathfrak{h}}_{j}{\mathsf{OPT}}_{\#}\leq\frac{f{\mathsf{OPT}}_{\#}}{\chi}$ , and therefore $\nicefrac{{1}}{{f}}\leq\frac{\mathcal{K}_{i}}{\mathcal{K}_{j}}\leq f$ for any two $i,j\in\{1,\dots,\chi\}$ . Let $\mathcal{E}_{j}$ be the event defined as $\mathcal{E}_{j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\mathcal{K}_{j}}{2f}\leq c_{j}\leq 5\mathcal{K}_{j}$ . Then for any two $i,j\in\{1,\dots,\chi\}$ we get

\Pr\left[{\textstyle\frac{p_{i}}{p_{j}}\leq 10f^{2}}\right]\geq\Pr\left[\mathcal{E}_{i}\wedge\mathcal{E}_{j}\right]=1-\Pr\left[\,\overline{\mathcal{E}_{i}}\vee\overline{\mathcal{E}_{j}}\,\right]\geq 1-\Pr\left[\,\overline{\mathcal{E}_{i}}\,\right]-\Pr\left[\,\overline{\mathcal{E}_{j}}\,\right]\\ \geq 1-\Pr\left[p_{i}>5\mathcal{K}_{i}\right]-\Pr\left[p_{i}<\nicefrac{{\mathcal{K}_{i}}}{{(2f)}}\right]-\Pr\left[p_{j}>5\mathcal{K}_{j}\right]-\Pr\left[p_{j}<\nicefrac{{\mathcal{K}_{j}}}{{(2f)}}\right]\\ >1-2\big{(}{\chi}^{-96}+{\chi}^{-2}\big{)}

(10)

The assumption of $\mathcal{K}_{j}\geq 16a\ln\chi$ , is satisfied provided ${\mathsf{OPT}}_{\#}\geq 16a\chi\ln\chi$ . (10) implies our claim in (d)(ii) using the technique in Section 9.3.

9.6 Alg-small-opt_#: details and proofs of relevant claims in (a)–(c) and (d)(iii)

Another modification of the ${\mathsf{LP}}$ -relaxation in Fig. 3

Note that for this case $\chi=O(1)$ . Fix an optimal solution for our instance of Fmc. Let $\mathcal{A}_{i}$ be the collection of those sets that contain at least one element of color $i$ and let $Z_{i}=\sum_{\mathcal{S}_{\ell}\in\mathcal{A}_{i}}y_{\ell}$ indicate the number of sets from $\mathcal{A}_{i}$ selected in an integral solution of the ${\mathsf{LP}}$ ; obviously $Z_{i}\geq 1$ . We consider two cases for $Z_{i}$ depending on whether it is at most $5\ln\chi$ or not. We cannot know a priori whether $Z_{i}\leq 5\ln\chi$ or not. However, for our analysis it suffices if we can guess just one set belonging to $Z_{i}$ correctly. We can do this by trying out all relevant possibilities exhaustively in the following manner. Let $\Psi=\{1,\dots,\chi\}$ be the set of indices of all colors. For each of the $2^{\Psi}-1$ subsets $\Psi^{\prime}$ of $\Psi$ , we “guess” that $Z_{i}\leq 5\ln\chi$ if and only if $i\in\Psi^{\prime}$ . Of course, we still do not know one set among these $5\ln\chi$ subset for each such $i$ , so we will exhaustively try out one each of the at most $|\mathcal{A}_{i}|\leq m$ sets for each $i$ . For every such choice of $\Psi^{\prime}$ and every such choice of a set $\mathcal{S}_{i_{\Psi^{\prime}}}\in\mathcal{A}_{i}$ for each $i\in\Psi^{\prime}$ , we perform the following steps:

$\triangleright$

Select the sets $\mathcal{S}_{i_{\Psi^{\prime}}}$ and their elements for each $i\in\Psi^{\prime}$ Set the variables corresponding to these sets and elements to $1$ in the ${\mathsf{LP}}$ -relaxation in Fig. 3, i.e., set $y_{i_{\Psi^{\prime}}}=1$ and $x_{j}=1$ for every $i\in\Psi^{\prime}\mathcal{S}_{i_{\Psi^{\prime}}}$ and $j\in$ . Remove any constraint that is already satisfied after the above step.
$\triangleright$

Add the following additional (at most $\chi$ ) constraints to the ${\mathsf{LP}}$ -relaxation:

$\sum_{(u_{\ell}\in C_{i})\wedge(u_{\ell}\in\mathcal{S}_{j})}y_{j}>5\ln\chi$ for $i\notin\Psi^{\prime}$

Note that the total number of iterations of the basic iterations that is needed is at most $O({(2m)}^{\chi})$ , which is polynomial provided $\chi=O\big{(}\max\big{\{}1,\,\frac{\log n}{\log m}\big{\}}\big{)}$ .

Analysis of the modified ${\mathsf{LP}}$ -relaxation

We now analyze that iteration of the ${\mathsf{LP}}$ -relaxation that correctly guesses the value of ${\mathsf{OPT}}_{\#}$ , the subset $\Psi^{\prime}\subseteq\Psi$ and the sets $\mathcal{S}_{i_{\Psi^{\prime}}}\in\mathcal{A}_{i}$ for each $i\in\Psi^{\prime}$ . As already mentioned elsewhere, the random variables $x_{1}^{+},\dots,x_{n}^{+}$ may not be pairwise independent since two distinct elements belonging to the same set are correlated, and consequently the random variables $p_{1},\dots,p_{\chi}$ also may not be pairwise independent. For convenience, let $\mu_{i}={\mathbb{E}}[x_{i}^{+}]$ and $\mathcal{E}_{i}$ denote the event $\mathcal{E}_{i}\equiv x_{i}^{+}=1$ ; note that $(1-{\mathbf{e}}^{-1})x_{i}^{\ast}<\Pr\left[\mathcal{E}_{i}\right]=\mu_{i}\leq\min\{1,\,2fx_{i}^{\ast}\}$ . We first calculate a bound on $\mbox{cov}(x_{i}^{+},x_{j}^{+})$ for all $i\neq j$ as follows. If $x_{i}^{+}$ and $x_{j}^{+}$ are independent then of course $\mbox{cov}(x_{i}^{+},x_{j}^{+})=0$ , otherwise

-\min\left\{\mu_{i},\,\mu_{j}\right\}\leq-\Pr\left[\mathcal{E}_{i}\right]\Pr\left[\mathcal{E}_{j}\right]\leq\Pr\left[\mathcal{E}_{i}\wedge\mathcal{E}_{j}\right]-\Pr\left[\mathcal{E}_{i}\right]\Pr\left[\mathcal{E}_{j}\right]={\mathbb{E}}[x_{i}^{+}x_{j}^{+}]-\mu_{i}\mu_{j}\\ =\mbox{cov}(x_{i}^{+},x_{j}^{+})\leq\Pr\left[\mathcal{E}_{i}\wedge\mathcal{E}_{j}\right]-\mu_{i}\mu_{j}\leq\min\left\{\mu_{i},\,\mu_{j}\right\}-\mu_{i}\mu_{j}<\min\left\{\mu_{i},\,\mu_{j}\right\}

giving the following bounds:

\displaystyle-\min\left\{\varrho(f)x_{i}^{\ast},\,\varrho(f)x_{j}^{\ast}\,\right\}\leq\mbox{cov}(x_{i}^{+},x_{j}^{+})\leq\min\left\{2fx_{i}^{\ast},\,2fx_{j}^{\ast},\,1\right\}

(11)

For notational convenience, let $\mathcal{D}_{i,j}=\{\ell\,|\,u_{i},u_{j}\in\mathcal{S}_{\ell},\,j\neq i\}$ be the indices of those sets in which both the elements $u_{i}$ and $u_{j}$ appear, and let $\mathcal{D}_{i}=\cup_{j=1}^{n}\mathcal{D}_{i,j}$ . Note that $|\mathcal{D}_{i,j}|\leq f$ , $|\mathcal{D}_{i}|\leq(a-1)f$ , and the random variable $x_{i}^{+}$ is independent of all $x_{j}^{+}$ satisfying $j\notin\mathcal{D}_{i}$ . Using this observation and (11), for any $i\in\{1,\dots,n\}$ we get

\displaystyle\sum_{j=1}^{n}\mbox{cov}(x_{i}^{+},x_{j}^{+})\leq\sum_{j\in\mathcal{D}_{i}}\!\!\big{(}\min\left\{\mu_{i},\,\mu_{j}\right\}\big{)}\leq|\mathcal{D}_{i}|\,\mu_{i}\leq af\mu_{i}\leq\min\big{\{}2af^{2}x_{i}^{\ast},\,af\big{\}}

(12)

\sum_{j=1}^{n}\mbox{cov}(x_{i}^{+},x_{j}^{+})\geq-\sum_{j\in\mathcal{D}_{i}}\min\left\{\mu_{i},\,\mu_{j}\right\}\geq-\min\Big{\{}|\mathcal{D}_{i}|\,\mu_{i},\,\varrho(f)\sum_{j\in\mathcal{D}_{i}}x_{j}^{\ast}\Big{\}}\\ \geq-\min\Big{\{}(a-1)f\,\mu_{i},\,\varrho(f)\sum_{j\in\mathcal{D}_{i}}x_{j}^{\ast}\Big{\}}>-afx_{i}^{\ast}

(13)

The above bounds can be used to bound the total pairwise co-variance between elements in two same or different color classes as follows. Consider two color classes $C_{i}$ and $C_{j}$ ( $i=j$ is allowed). Then,

\sum_{u_{r}\in C_{i}}\sum_{u_{s}\in C_{j}}\mbox{cov}(x_{r}^{+},x_{s}^{+})\leq\sum_{u_{r}\in C_{i}}\sum_{j=1}^{n}\mbox{cov}(x_{r}^{+},x_{j}^{+})\leq\sum_{u_{r}\in C_{i}}\min\big{\{}2af^{2}x_{r}^{\ast},\,af\big{\}}\\ \textstyle=\min\Big{\{}2af^{2}\sum_{u_{r}\in C_{i}}x_{r}^{\ast},\,af|C_{i}|\Big{\}}\leq\min\big{\{}2af^{2}\frac{{\mathsf{OPT}}_{\#}}{\chi},\,afn\big{\}}

(14)

\displaystyle\sum_{u_{r}\in C_{i}}\sum_{u_{s}\in C_{j}}\mbox{cov}(x_{r}^{+},x_{s}^{+})\geq-\sum_{u_{r}\in C_{i}}\sum_{j\in\mathcal{D}_{r}}\min\{\mu_{r},\,\mu_{j}\}>-\sum_{u_{r}\in C_{i}}afx_{r}^{\ast}=\textstyle-af\frac{{\mathsf{OPT}}_{\#}}{\chi}

(15)

For calculations of probabilities of events of the form “ $p_{i}>\Delta p_{j}$ ”, we first need to bound the probability of events “ $p_{j}=0$ ” for $j\in\{1,\dots,\chi\}$ . If $j\in\Psi^{\prime}$ then $\Pr\left[p_{j}=0\right]=0$ since at least one set containing an element of color $j$ is always selected. Otherwise, $\sum_{\mathcal{S}_{\ell}\in\mathcal{A}_{j}}y_{\ell}^{\ast}>5\ln\chi$ , and $p_{j}=0$ if and only if $y_{\ell}^{+}=0$ for every $\mathcal{S}_{\ell}\in\mathcal{A}_{j}$ . This gives us the following bound for $j\notin\Psi^{\prime}$ :

\Pr\left[p_{j}=0\right]=\prod_{\mathcal{S}_{\ell}\in\mathcal{A}_{j}}\Pr\left[y_{\ell}^{+}=0\right]=\prod_{\mathcal{S}_{\ell}\in\mathcal{A}_{j}}(1-y_{\ell}^{\ast})\leq\prod_{\mathcal{S}_{\ell}\in\mathcal{A}_{j}}{\mathbf{e}}^{-y_{\ell}^{\ast}}\\ ={\mathbf{e}}^{-\sum_{\mathcal{S}_{\ell}\in\mathcal{A}_{j}}y_{\ell}^{\ast}}\leq{\mathbf{e}}^{-5\ln\chi}=\chi^{-5}

Combining both cases, we have $\Pr\left[p_{j}=0\right]\leq\nicefrac{{1}}{{\chi^{5}}}$ for all $j$ .

We now can calculate the probabilities of events of the form “ $p_{i}>\Delta p_{j}$ ” for $\Delta=\Delta_{1}+\Delta_{2}\geq 1$ , $\Delta_{1},\Delta_{2}\geq 0$ and $i,j\in\{1,\dots,\chi\}$ as follows:

\Pr\left[p_{i}\geq\Delta p_{j}\right]\leq\Pr\left[p_{j}=0\right]+\Pr\left[p_{i}\geq\Delta p_{j}\,|\,p_{j}\geq 1\right]\leq\chi^{-5}+\Pr\left[p_{i}\geq\Delta_{1}p_{j}+\Delta_{2}\,|\,p_{j}\geq 1\right]\\ =\chi^{-5}+\frac{\Pr\left[(p_{i}\geq\Delta_{1}p_{j}+\Delta_{2})\wedge(p_{j}\geq 1)\right]}{\Pr\left[p_{j}\geq 1\right]}<\chi^{-5}+\frac{\Pr\left[p_{i}\geq\Delta_{1}p_{j}+\Delta_{2}\right]}{1-\Pr\left[p_{j}=0\right]}\\ <\chi^{-5}+\frac{\Pr\left[p_{i}\geq\Delta_{1}p_{j}+\Delta_{2}\right]}{1-\chi^{-5}}

(16)

For a real number $\zeta>0$ , let $\delta_{i,j}=p_{i}-\zeta p_{j}$ . We have the following bound on ${\mathbb{E}}[\delta_{i,j}]$ for all $\zeta\geq 3f$ :

\displaystyle{\mathbb{E}}[\delta_{i,j}]={\mathbb{E}}[p_{i}]-\zeta\,{\mathbb{E}}[p_{j}]\leq 2f\frac{{\mathsf{OPT}}_{\#}}{\chi}-\zeta\varrho(f)\frac{{\mathsf{OPT}}_{\#}}{\chi}<0

Therefore, using Chebyshev’s inequality we get (for all $\zeta\geq 3f$ and $\lambda>1$ ):

\Pr\left[p_{i}\geq\zeta p_{j}+\lambda\sqrt{\mbox{var}(\delta_{i,j})}\right]=\Pr\left[\delta_{i,j}\geq\lambda\sqrt{\mbox{var}(\delta_{i,j})}\right]\\ <\Pr\left[\left|\,\delta_{i,j}-{\mathbb{E}}[\delta_{i,j}]\,\right|>\lambda\sqrt{\mbox{var}(\delta_{i,j})}\right]\leq\nicefrac{{1}}{{\lambda^{2}}}

(17)

Using (17) in (16) with $\Delta_{1}=\zeta$ , $\Delta_{2}=\lambda\sqrt{\mbox{var}(\delta_{i,j})}$ and $\lambda=10\chi$ we get

\textstyle\Pr\left[p_{i}<\left(\zeta+10\chi\sqrt{\mbox{var}(\delta_{i,j})}\,\right)p_{j}\right]=1-\Pr\left[p_{i}\geq\left(\zeta+10\chi\sqrt{\mbox{var}(\delta_{i,j})}\,\right)p_{j}\right]\\ \textstyle>1-\chi^{-5}-\frac{\chi^{-100}}{1-\chi^{-5}}>1-\chi^{-4}

(18)

We now calculate a bound on $\mbox{var}(\delta_{i,j})$ using (14) and (15) as follows:

\mbox{var}(\delta_{i,j})=\mbox{var}(p_{i}-\zeta p_{j})=\mbox{var}\Big{(}\sum_{u_{\ell}\in C_{i}}\!\!x_{\ell}^{+}+\sum_{u_{\ell}\in C_{j}}\!\!(-\zeta x_{\ell}^{+})\Big{)}\\ =\sum_{u_{\ell}\in C_{i}}\!\!\mbox{var}\big{(}x_{\ell}^{+}\big{)}+\zeta^{2}\sum_{u_{\ell}\in C_{j}}\!\!\mbox{var}\big{(}x_{\ell}^{+}\big{)}+\!\!\!\!\!\!\sum_{u_{r},u_{s}\in C_{i},r\neq s}\!\!\!\!\!\!\!\!\!\mbox{cov}(x_{r}^{+},x_{s}^{+})+\!\!\!\!\!\!\sum_{u_{r},u_{s}\in C_{j},r\neq s}\!\!\!\!\!\!\!\!\!\mbox{cov}(-\zeta x_{r}^{+},-\zeta x_{s}^{+})\\ +\!\!\!\!\!\!\sum_{u_{r}\in C_{i},u_{s}\in C_{j}}\!\!\!\!\!\!\!\!\!\mbox{cov}(x_{r}^{+},-\zeta x_{s}^{+})\\ \leq\sum_{u_{\ell}\in C_{i}}\!\!\mu_{\ell}+\zeta^{2}\sum_{u_{\ell}\in C_{j}}\!\!\mu_{\ell}+2af^{2}{\textstyle\frac{{\mathsf{OPT}}_{\#}}{\chi}}+\zeta^{2}\!\!\!\!\!\!\sum_{u_{r},u_{s}\in C_{j},r\neq s}\!\!\!\!\!\!\!\!\!\mbox{cov}(x_{r}^{+},x_{s}^{+})-\zeta\!\!\!\!\!\!\sum_{u_{r}\in C_{i},u_{s}\in C_{j}}\!\!\!\!\!\!\!\!\!\mbox{cov}(x_{r}^{+},x_{s}^{+})\\ \leq{\mathbb{E}}[c_{i}]+\zeta^{2}{\mathbb{E}}[c_{j}]+2af^{2}{\textstyle\frac{{\mathsf{OPT}}_{\#}}{\chi}}+2\zeta^{2}af^{2}{\textstyle\frac{{\mathsf{OPT}}_{\#}}{\chi}}+\zeta af{\textstyle\frac{{\mathsf{OPT}}_{\#}}{\chi}}\\ \leq 2f{\textstyle\frac{{\mathsf{OPT}}_{\#}}{\chi}}+\zeta^{2}2f{\textstyle\frac{{\mathsf{OPT}}_{\#}}{\chi}}+2af^{2}{\textstyle\frac{{\mathsf{OPT}}_{\#}}{\chi}}+\zeta^{2}2af^{2}{\textstyle\frac{{\mathsf{OPT}}_{\#}}{\chi}}+\zeta af{\textstyle\frac{{\mathsf{OPT}}_{\#}}{\chi}}\leq 4\zeta^{2}af^{2}{\textstyle\frac{{\mathsf{OPT}}_{\#}}{\chi}}\\ \Rightarrow\,\sqrt{\mbox{var}(\delta_{i,j})}\leq 2\zeta\sqrt{a}f\sqrt{{\mathsf{OPT}}_{\#}/\chi}

(19)

Setting $\zeta=3f$ and using (19) in (18) we get $\Pr\left[c_{i}<\left(3f+60\sqrt{a}f^{2}\sqrt{{\mathsf{OPT}}_{\#}\chi}\,\right)c_{j}\right]>1-\chi^{-4}$ . This implies our claim in (d)(iii) using the technique in Section 9.3.

9.7 Limitations of our ${\mathsf{LP}}$ -relaxation: “a gap of factor $f$ ” for coloring constraints

The coloring constraint bounds in Theorem 9.1(e)(i)–(ii) depend on $f$ or $f^{2}$ only. It is natural to ask as a possible first direction of improvement whether this dependence can be eliminated or improved by better analysis of our ${\mathsf{LP}}$ -relaxations. Proposition 1 shows that this may not be possible even for $\chi=2$ unless one uses a significantly different ${\mathsf{LP}}$ -relaxation for Fmc( $\chi,k$ ).

Proposition 1

There exists optimal non-integral solutions of Fmc with the following property: any rounding approach that does not change the values of zero-valued variables in the fractional solution must necessarily result in an integral solutions in which the color constraints differ by at least a factor of $f$ .

Proof

We will show our result for the ${\mathsf{LP}}$ -relaxation in Fig. 3; proofs for other modified versions of this ${\mathsf{LP}}$ -relaxation are similar. Consider $\alpha\gg 1$ disjoint collections of sets and elements of the following type: for $j\in\{1,\dots,\alpha\}$ , the $j^{\mathrm{th}}$ collection consists of a set of $\alpha+1$ elements $\mathcal{U}^{j}=\{u_{1}^{j},\dots,u_{\alpha+1}^{j}\}$ with $\mathcal{C}(u_{1}^{j})=1$ and $\mathcal{C}(u_{2}^{j})=\dots=\mathcal{C}(u_{\alpha+1}^{j})=2$ , and the $\alpha+1$ sets $\mathcal{S}_{1}^{j},\dots,\mathcal{S}_{\alpha+1}^{j}$ where $\mathcal{S}_{i}^{j}=\mathcal{U}^{j}\setminus\{u_{i}^{j}\}$ for $i\in\{1,\dots,\alpha+1\}$ (note that each element $u_{i}^{j}$ is in exactly $\alpha$ sets). Add to these collections the additional $2\alpha+2$ elements $u_{1}^{\ell},u_{2}^{\ell}$ with $\mathcal{C}(u_{1}^{\ell})=1$ and $\mathcal{C}(u_{2}^{\ell})=2$ , and the $\alpha+1$ sets $\mathcal{S}^{\ell}=\{u_{1}^{\ell},u_{2}^{\ell}\}$ for $\ell\in\{\alpha+1,\dots,2\alpha+1\}$ . Note that for our created instance $f=\alpha$ . Consider the following two different solutions of the ${\mathsf{LP}}$ -relaxation:

(1): For a non-integral solution, let $y_{1}^{j}=\dots=y_{\alpha+1}^{j}=\nicefrac{{1}}{{\alpha}}$ , let $x_{1}^{j}=1$ and let $x_{2}^{j}=\dots=x_{\alpha+1}^{j}=\nicefrac{{1}}{{\alpha}}$ for $j\in\{1,\dots,\alpha\}$ , and set all other variables to zero. This results in a solution with summation of set variables being $\alpha+1$ (i.e., $\alpha+1$ sets are selected non-integrally), and summation of element variables being $2\alpha+2$ (i.e., $2\alpha+2$ elements are selected non-integrally). Moreover, the summation of element variables with the color value of $1$ is precisely the same as summation of element variables with the color value of $2$ since both are equal to $\alpha+1$ .
(2): For an integral solution, let $y^{\ell}=x_{1}^{\ell}=x_{2}^{\ell}=1$ for $\ell\in\{\alpha+1,\dots,2\alpha+1\}$ . This also results in a solution in which $\alpha+1$ sets are selected, the number of elements covered is $2\alpha+2$ and the number of elements of each color is $\alpha+1$ .

The crucial things to note here is that the two above solutions are disjoint (i.e., non-zero variables in one solution are zero in the other and vice versa), and thus any rounding approach for the solution in (1) that does not change values of the zero-valued variables results in an integral solution in which the number of elements of color $2$ is $f$ times the number of elements of color $1$ .

10 A tale of fewer colors: deterministic approximation for Fmc when $\chi$ is “not too large”

In this section we provide polynomial-time deterministic approximations of Fmc via the iterated rounding technique for ${\mathsf{LP}}$ -relaxations. We assume that the reader is familiar with the basic concepts related to this approach as described, for example, in LRS11 . Our approximation qualities will depend on the parameters $f$ and $\chi$ and the coloring constraint bounds are interesting only if $\chi$ is not too large, e.g., no more than, say, poly-logarithmic in $n$ . For better understanding of the idea, we will first consider the special case Node-fmc of Fmc for which $f=2$ , and later on describe how to adopt the same approach for arbitrary $f$ . As per the proof of Theorem 9.1 (see Section 9.2) we may assume we know the value of ${\mathsf{OPT}_{\#}}$ exactly. A main ingredient of the iterated rounding approach is the following “rank lemma”.

Fact 2 (Rank lemma)

(LRS11, , Lemma 2.1.4) Consider any convex polytope $P\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{\mathbf{x}\in{\mathbb{R}}^{n}\,|\,\mathbf{A}_{j}\mathbf{x}\geq b_{j}\text{ for }j\in\{1,\dots,m\},\,\mathbf{x}\geq\mathbf{0}\}$ for some $\mathbf{A}_{1},\dots,\mathbf{A}_{m}\in{\mathbb{R}}^{n}$ and $(b_{1},\dots,b_{m})^{\mathrm{T}}\in{\mathbb{R}}^{m}$ . Then the following property holds for every extreme-point for $P$ : the number of any maximal set of linearly independent tight constraints $($ i.e., constraints satisfying $\mathbf{A}_{j}\mathbf{x}=b_{j}$ for some $j)$ in this solution equals the number of non-zero variables.

10.1 Approximating Node-fmc

Theorem 10.1

We can design a deterministic polynomial-time approximation algorithm Alg-iter-round for Node-fmc with the following properties:

(a)

The algorithm selects $\tau$ nodes where $\tau\leq\begin{cases}k+\frac{\chi-1}{2},&\mbox{if $\chi=O(1)$}\\ k+\chi-1,&\mbox{otherwise}\end{cases}$
(b)

The algorithm is a $\frac{1}{2}$ -approximation for Node-fmc, i.e., the total weight of the selected elements is at least $\nicefrac{{{\mathsf{OPT}}}}{{2}}$ .
(c)

The algorithm satisfies the $\varepsilon$ -approximate coloring constraints $($ cf. Inequality (5) $)$ as follows:

for all $i,j\in\{1,\dots,\chi\}$ , $\frac{p_{i}}{p_{j}}<\begin{cases}4+4\chi,&\mbox{if $\chi=O(1)$}\\ 4+2\chi+4\chi^{2},&\mbox{otherwise}\end{cases}$

We discuss the proof in the rest of this section. Let $G=(V,E)$ be the given graph, and let $\mathsf{deg}(v)$ denote the degree of node $v$ . Assume that $G$ has no isolated nodes.

10.1.1 The case of $\chi=O(1)$

Since the problem can be exactly solved in polynomial time by exhaustive enumeration if $k$ is a constant, we can assume $k$ is at least a sufficiently large integer, e.g., assume that $k>10\chi$ .

Initial preprocessing

To begin, we “guess” $\chi+1$ nodes, say $v_{1},\dots,v_{\chi+1}\in V$ with $\mathsf{deg}(v_{1})\leq\mathsf{deg}(v_{2})\leq\dots\leq\mathsf{deg}(v_{\chi+1})$ such that there exists an optimal solution contains these $\chi+1$ nodes with the following property: “the remaining $k-(\chi+1)$ nodes in the solution have degree at most $\mathsf{deg}(v_{1})$ ”. Since there are at most $\binom{n}{\chi+1}=n^{O(1)}$ choices for such $\chi+1$ nodes, we can try them out in an exhaustive fashion. Thus, we only need to analyze that run of our algorithm where the our guess is correct. Once these $\chi+1$ nodes have been selected, we will use the following sets of nodes in $\widehat{V}$ and ”incidence-indexed” edges $\widehat{E}$ as input to our algorithm (note that an edge $e\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{u,v\}$ may appear as two members $(e,u)$ and $(e,v)$ in $\widehat{E}$ if both $u$ and $v$ are in $\widehat{V}$ ):

	$\displaystyle\widehat{V}=V\setminus\big{(}\,\{v_{1},\dots,v_{\chi+1}\}\,\cup\,\{v\,\|\,\text{degree of $v$ in $G$ is strictly larger than $\mathsf{deg}(v_{1})$ }\}\,\big{)}$
	$\displaystyle\widehat{E}=\{\,(e,u)\,\|\,\big{(}e\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{u,v\}\in E\big{)}\bigwedge\big{(}v\notin\{v_{1},\dots,v_{\chi+1}\}\big{)}\bigwedge\big{(}u\in\widehat{V}\big{)}\,\}$

Fix an optimal solution $V_{\mathrm{opt}}\subseteq V$ that includes the nodes $v_{1},\dots,v_{\chi+1}$ . We next make the following parameter adjustments:

$\triangleright$

We update an estimate for $p_{j}$ (the number of edges of color $j$ covered by the optimal solution) from its initial value of $\nicefrac{{{\mathsf{OPT}}_{\#}}}{{\chi}}$ in the following manner. Let $\mu_{j}$ be the number of edges of color $j$ incident on at least one of the nodes in $\{v_{1},\dots,v_{\chi+1}\}$ . Consider the quantity ${\widehat{q}}_{j}=\sum_{u\in V_{\mathrm{opt}}\setminus\{v_{1},\dots,v_{\chi+1}\}}\big{|}\,\big{\{}e\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{u,v\}\in E\,|\,(v\notin\{v_{1},\dots,v_{\chi+1}\})\,\wedge\,(\mathcal{C}(e)=j)\big{\}}\,\big{|}$ . Note that $\sum_{i=1}^{\chi+1}\mathsf{deg}(v_{i})\leq 2{\mathsf{OPT}_{\#}}$ and ${\widehat{q}}_{j}$ is an integer in the set $\Big{\{}\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{j},\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{j}+1,\dots,2\big{(}\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{j}\big{)}\Big{\}}\subseteq\{0,1,2,\dots,n\}$ since any edge can be covered by either one or two nodes. Note that there are at most $\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{j}+1\leq\frac{{\mathsf{OPT}_{\#}}}{\chi}\leq\frac{n}{\chi}$ possible number of integers values that each ${\widehat{q}}_{j}$ may take. Since $\chi=O(1)$ , we can try out all possible combinations of ${\widehat{q}}_{j}$ values over all colors in polynomial time since $(n/\chi)^{\chi}=n^{O(1)}$ . Thus, we henceforth assume that we know the correct value of ${\widehat{q}}_{j}$ for each $j\in\{1,\dots,\chi\}$ . Note that ${\widehat{q}}_{j}\geq 0$ since our guess is correct.
$\triangleright$

Update $k$ (the number of nodes to be selected) by subtracting $\chi+1$ from it, and call the new value $\widehat{k}$ .
$\triangleright$

Update $C_{i}$ (the set of edges of color $i$ ) to be the set of edges in $\widehat{E}$ that are of color $i$ .

Yet another ${\mathsf{LP}}$ -relaxation

Let $|\widehat{V}|=\widehat{n}$ and $|\widehat{E}|=\widehat{m}$ . We will start with an initial ${\mathsf{LP}}$ -relaxation of Node-fmc on $\widehat{G}$ which will be iteratively modified by our rounding approach. Our ${\mathsf{LP}}$ -relaxation is the following modified version of the ${\mathsf{LP}}$ -relaxation in Fig. 3.

$\triangleright$

There is a node indicator variable $y_{v}$ for every node $v\in\widehat{V}$ and an edge indicator variable $x_{e,u}$ for every edge $(e,u)\in\widehat{E}$ ; thus we have $\widehat{n}+\widehat{m}$ variables in total.
$\triangleright$

Constraints of the form “ $x_{j}\geq y_{\ell}$ ” and “ $x_{j}\leq\sum_{u_{j}\in\mathcal{S}_{\ell}}y_{\ell}$ ” in Fig. 3 are removed now and instead replaced by at most two constraints $x_{e,u}=y_{u}$ if $y_{u}\in\widehat{V}$ and $x_{e,v}=y_{v}$ if $y_{v}\in\widehat{V}$ . This is done so that we can apply the rank lemma in a meaningful way.
$\triangleright$

Note that the quantity $\sum_{u\in\widehat{V}}\sum_{e\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{u,v\}\in C_{i}}x_{e,u}$ for each color $i$ is the integer ${\widehat{q}}_{i}$ mentioned before.
$\triangleright$

To maximize the parameter ranges over which our algorithm can be applied, we replace the $\binom{\chi}{2}$ constraints in Fig. 3 of the form “ $\sum_{u_{\ell}\in C_{i}}x_{\ell}=\sum_{u_{\ell}\in C_{j}}x_{\ell}$ for $i,j\in\{1,\dots,\chi\}$ , $i<j$ ” by the $\chi$ constraints $\sum_{u\in\widehat{V}}\sum_{e\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{u,v\}\in C_{i}}x_{e,u}=q_{i}^{\prime}$ for $i\in\{1,\dots,\chi\}$ .

The entire initial ${\mathsf{LP}}$ -relaxation $\mathcal{L}$ for $\widehat{G}$ is shown in Fig. 5 for convenience. Note that the number of constraints in lines (1)–(3) of Fig. 5 is exactly $\widehat{m}+\chi+1$ .

	maximize	$\psi=\sum_{u\in\widehat{V}}\sum_{(e,u)\in\widehat{E}}w(e)x_{e,u}$
	subject to
(1)		$x_{e,u}=y_{u}$	for all $u\in\widehat{V}$ and $(e,u)\in\widehat{E}$
(2)		$\sum_{v\in V_{1}}y_{v}=\widehat{k}$
(3)		$\sum_{u\in\widehat{V}}\sum_{e\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{u,v\}\in C_{i}}x_{e,u}={\widehat{q}}_{i}$	for all $i\in\{1,\dots,\chi\}$
(4)		$0\leq x_{e,u}\leq 1$	for all $(e,u)\in\widehat{E}$
(5)		$0\leq y_{v}\leq 1$	for all $v\in\widehat{V}$

Figure 5: The initial

{\mathsf{LP}}

-relaxation

\mathcal{L}=\mathcal{L}^{(0)}

for the graph

\widehat{G}

used in Theorem 10.1. The iterated rounding approach will successively modify the

{\mathsf{LP}}

to create a sequence

\mathcal{L}^{(1)},\mathcal{L}^{(2)},\dots

{\mathsf{LP}}

’s.

Details of iterated rounding

We will use the variable $t\in\{0,1,2,\dots,n\}$ to denote the iteration number of our rounding, with $t=0$ being the situation before any rounding has been performed, and we will use a “superscript $(t)$ ” for the relevant quantities to indicate their values or status after the $t^{\mathrm{th}}$ iteration of the rounding, e.g., $\widehat{n}^{(0)}=\widehat{n}$ and $\widehat{n}^{(1)}$ is the value of $\widehat{n}$ after the first iteration of rounding. Our iterated rounding algorithm Alg-iter-round in high level details is shown in Fig. 6, where the following notation is used for brevity for a node $u\in\widehat{V}$ :

\displaystyle Z_{u}^{\mathrm{variables}}=\{y_{u}\}\,\bigcup\,\big{\{}\,x_{e,u}\,|\,(e,u)\in\widehat{E}\big{\}}

For concise analysis of our algorithm, we will use the following notations:

$\triangleright$

$W^{\chi+1}$ is the sum of weights of all the edges incident to one or more nodes from the set of nodes $\{v_{1},\dots,v_{\chi+1}\}$ .
$\triangleright$

$w(X)=\sum_{x_{u,e}\in X}w(e)$ for a subset of variable $X\subseteq\{x_{e,u}\,|\,(e,u)\in\widehat{E}\}$ .
$\triangleright$

$W_{\!\!\text{\tiny\sc Alg}}^{(t)}=w(\widehat{X}_{\mathrm{sol}}^{(t)})$ is the sum of weights of the edges whose variables are in $\widehat{X}_{\mathrm{sol}}^{(t)}$ (thus, for example, $W_{\!\!\text{\tiny\sc Alg}}^{(0)}=0$ ).
$\triangleright$

${\mathsf{OPT}}_{\mathrm{frac}}^{(t)}$ is the optimum value of the objective function of the ${\mathsf{LP}}$ -relaxation $\mathcal{L}^{(t)}$ during the $t^{\mathrm{th}}$ iteration of rounding.
$\triangleright$

${\widehat{p}}_{i}^{(t)}$ is the number of edges of color $i$ selected by Alg-iter-round up to and including the $t^{\mathrm{th}}$ iteration of rounding.
$\triangleright$

$t^{\mathrm{final}}$ is the value of $t$ in the last iteration of rounding.

(*\text{ initialization }*)

t\leftarrow 0

\widehat{V}_{\mathrm{sol}}\leftarrow\emptyset

;

\widehat{X}_{\mathrm{sol}}\leftarrow\emptyset

;

\mbox{var-count}_{\mathrm{remaining}}\leftarrow\widehat{n}+\widehat{m}

;

\widehat{r}\leftarrow\widehat{m}

;

\widehat{n}\leftarrow\widehat{n}

;

(*\text{ iterations of rounding }*)

while

(\mbox{var-count}_{\mathrm{remaining}}\neq 0)

t\leftarrow t+1

find an extreme-point optimal solution of objective value

{\mathsf{OPT}}_{\mathrm{frac}}^{(t-1)}

for the

{\mathsf{LP}}

\mathcal{L}^{(t-1)}

(cf. Fig. 5)

begin cases

Case 1: there exists a variable

y_{u}

in the solution such that

y_{u}=0

\mbox{var-count}_{\mathrm{remaining}}\leftarrow\mbox{var-count}_{\mathrm{remaining}}-\big{|}\,Z_{u}^{\mathrm{variables}}\,\big{|}

;

\,\,\,\widehat{n}\leftarrow\widehat{n}-1

\widehat{r}\leftarrow\widehat{r}-\big{|}\,\{\,x_{e,u}\,|\,x_{e,u}\in Z_{u}^{\mathrm{variables}}\,\}\,\big{|}

;

\,\,\,\widehat{X}_{\mathrm{sol}}\leftarrow\widehat{X}_{\mathrm{sol}}\cup\{\,x_{e,u}\,|\,x_{e,u}\in Z_{u}^{\mathrm{variables}}\,\}

remove the variables in

Z_{u}^{\mathrm{variables}}

from

\mathcal{L}^{(t-1)}

, and delete or update the constraints

and the objective function to reflect the removal of variables

Case 2: there exists a variable

y_{u}

in the solution such that

y_{u}=1

\widehat{V}_{\mathrm{sol}}\leftarrow\widehat{V}_{\mathrm{sol}}\cup\{u\}

;

\,\,\,\mbox{var-count}_{\mathrm{remaining}}\leftarrow\mbox{var-count}_{\mathrm{remaining}}-\big{|}\,Z_{u}^{\mathrm{variables}}\,\big{|}

\widehat{k}\leftarrow\widehat{k}-1

;

\,\,\,\widehat{n}\leftarrow\widehat{n}-1

;

\,\,\,\widehat{r}\leftarrow\widehat{r}-\big{|}\,\{\,x_{e,u}\,|\,x_{e,u}\in Z_{u}^{\mathrm{variables}}\,\}\,\big{|}

\widehat{X}_{\mathrm{sol}}\leftarrow\widehat{X}_{\mathrm{sol}}\cup\{\,x_{e,u}\,|\,x_{e,u}\in Z_{u}^{\mathrm{variables}}\,\}

remove the variables in

Z_{u}^{\mathrm{variables}}

from

\mathcal{L}^{(t-1)}

, and delete or update the constraints

and the objective function to reflect the removal of variables

\forall\,i\in\{1,\dots,\chi\}:\,{\widehat{q}}_{i}\leftarrow{\widehat{q}}_{i}-\big{|}\,\{\,x_{e,u}\,|\,x_{e,u}\in Z_{u}^{\mathrm{variables}}\text{ and }\mathcal{C}(e)=i\,\}\,\big{|}

subtract the value

\sum_{x_{e,u}\in Z_{u}^{\mathrm{variables}}}w(e)x_{e,u}

from the objective function

\psi^{(t-1)}

Case 3:

1\leq\widehat{n}\leq\chi+1

let

y_{u_{1}},\dots,y_{u_{\widehat{n}^{\prime}}}

be the remaining non-zero

1\leq\widehat{n}^{\prime}\leq\nicefrac{{\widehat{n}}}{{2}}

node indicator variables

\mbox{var-count}_{\mathrm{remaining}}\leftarrow 0

;

\,\,\,\widehat{k}\leftarrow 0

;

\,\,\,\widehat{n}\leftarrow 0

;

\,\,\,\widehat{r}\leftarrow 0

\forall\,i\in\{1,\dots,\chi\}:\,{\widehat{q}}_{i}\leftarrow{\widehat{q}}_{i}-\big{|}\,\{\,x_{e,u}\,|\,x_{e,u}\in Z_{u}^{\mathrm{variables}}\text{ and }\mathcal{C}(e)=i\,\}\,\big{|}

\widehat{V}_{\mathrm{sol}}\leftarrow\widehat{V}_{\mathrm{sol}}\cup\{u_{1},\dots,u_{\widehat{n}^{\prime}}\}

;

\,\,\,\widehat{X}_{\mathrm{sol}}\leftarrow\widehat{X}_{\mathrm{sol}}\cup\{\,x_{e,u_{j}}\,|\,x_{e,u_{j}}\in Z_{u_{j}}^{\mathrm{variables}},\,j\in\{1,\dots,\widehat{n}^{\prime}\}\,\}

end cases

end while

{\mathsf{OPT}}_{\mathrm{frac}}\leftarrow 0

return

\widehat{V}_{\mathrm{sol}}

Figure 6: Pseudo-code of the iterated rounding algorithm Alg-iter-round used in Theorem 10.1.

\widehat{V}_{\mathrm{sol}}^{(t^{\mathrm{final}})}

is the set of nodes selected in our solution.

Lemma 2

Alg-iter-round terminates after at most $n$ iterations and selects at most $k+\frac{\chi-1}{2}$ nodes.

Proof

For finite termination, it suffices to show that at least one of the three cases in Alg-iter-round always applies. Consider the first iteration, say when $t=\alpha$ , when neither Case 1 nor Case 2 applies. Note that this also implies that $x_{e,u}\notin\{0,1\}$ for any variable $x_{e,u}$ in ${\mathsf{LP}}^{(\alpha)}$ since otherwise the variable $y_{u}$ in ${\mathsf{LP}}^{(\alpha)}$ will be either $0$ or $1$ via the equality constraint $y_{u}=x_{e,u}$ and one of Case 1 or Case 2 will apply. Thus the total number of non-zero variables is $\widehat{n}^{(\alpha)}+\widehat{r}^{(\alpha)}$ . Since the constraints in lines (4)–(5) of Fig. 5 are not strict constraints now (i.e., not satisfied with equalities), the total number of any maximal set of strict constraints is at most the total number of constraints in lines (1)–(3) of Fig. 5, i.e., at most $\widehat{r}^{(\alpha)}+\chi+1$ . By the rank lemma (Fact 2) $\widehat{r}^{(\alpha)}+\chi+1\geq\widehat{n}^{(\alpha)}+\widehat{r}^{(\alpha)}\equiv\widehat{n}^{(\alpha)}\leq\chi+1$ , which implies Case 3 applies and the algorithm terminates.

We now prove the bound on the number of selected sets. The value of $\widehat{k}$ decreases by $1$ every time a new node is selected in Case 2 and remains unchanged in Case 1 where no node is selected. In the very last iteration involving Case 3, since $G$ has no isolated nodes the number of node indicator variables is at least the number of edge indicator variables, implying $\widehat{n}^{\prime}\leq\nicefrac{{\widehat{n}}}{{2}}\leq\frac{\chi+1}{2}$ . Since $\widehat{k}^{(t^{\mathrm{final}}-1)}\geq 1$ , the total number of nodes selected is at most $k+(\widehat{n}^{\prime}-1)\leq k+\frac{\chi-1}{2}$ .

Lemma 3

The sum of weights $\Gamma$ of the edges selected by Alg-iter-round is at least $\nicefrac{{{\mathsf{OPT}}}}{{2}}$ .

Proof

Let $W_{\!\!\text{\footnotesize\sc Alg}\!-W}^{(t)}=W_{\!\!\text{\tiny\sc Alg}}^{(t)}-W^{\chi+1}$ , and ${\mathsf{OPT}}_{\!-W}={\mathsf{OPT}}-W^{\chi+1}$ . The proof of Lemma 2 shows that Case 3 of Alg-iter-round is executed only when $t=t^{\mathrm{final}}$ . Thus, the details of Alg-iter-round in Fig. 6 imply the following sequence of assertions:

(i)

${\mathsf{OPT}}_{\mathrm{frac}}^{(0)}\geq{\mathsf{OPT}}_{\!-W}$ and ${\mathsf{OPT}}_{\mathrm{frac}}^{(t)}={\mathsf{OPT}}_{\mathrm{frac}}^{(t-1)}-\big{(}w(\widehat{X}_{\mathrm{sol}}^{(t)})-w(\widehat{X}_{\mathrm{sol}}^{(t-1)})\big{)}$ for $t\in\{1,\dots,t^{\mathrm{final}}-1\}$ . Since the variables $x_{e_{u,j}}\in\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}})}\setminus\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}}-1)}$ are at most $1$ , we have

{\mathsf{OPT}}_{\mathrm{frac}}^{(t^{\mathrm{final}})}={\mathsf{OPT}}_{\mathrm{frac}}^{(t^{\mathrm{final}}-1)}-\sum_{x_{e_{u,j}}\in\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}})}\setminus\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}}-1)}}w(e)x_{e_{u,j}}\geq{\mathsf{OPT}}_{\mathrm{frac}}^{(t^{\mathrm{final}}-1)}-\sum_{x_{e_{u,j}}\in\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}})}\setminus\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}}-1)}}w(e)\\ ={\mathsf{OPT}}_{\mathrm{frac}}^{(t^{\mathrm{final}}-1)}-\big{(}w(\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}})})-w(\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}}-1)})\big{)}

Using the fact that ${\mathsf{OPT}}_{\mathrm{frac}}^{(t^{\mathrm{final}})}=0$ , we can therefore unravel the recurrence to get

\displaystyle{\mathsf{OPT}}_{\mathrm{frac}}^{(t^{\mathrm{final}})}\geq{\mathsf{OPT}}_{\mathrm{frac}}^{(0)}-w(\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}})})\,\Rightarrow\,w(\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}})})\geq{\mathsf{OPT}}_{\mathrm{frac}}^{(0)}\geq{\mathsf{OPT}}_{\!-W}

(20)

(ii)

$W_{\!\!\text{\footnotesize\sc Alg}\!-W}^{(0)}=0$ and $W_{\!\!\text{\footnotesize\sc Alg}\!-W}^{(t)}=W_{\!\!\text{\footnotesize\sc Alg}\!-W}^{(t-1)}+\big{(}w(\widehat{X}_{\mathrm{sol}}^{(t)})-w(\widehat{X}_{\mathrm{sol}}^{(t-1)})\big{)}$ for $t\in\{1,\dots,t^{\mathrm{final}}\}$ . Using (20) we can unravel the recurrence we get

\displaystyle W_{\!\!\text{\footnotesize\sc Alg}\!-W}^{(t^{\mathrm{final}})}=w(\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}})})\geq{\mathsf{OPT}}_{\mathrm{frac}}^{(0)}\geq{\mathsf{OPT}}_{\!-W}

Noting that an edge $e\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{u,v\}$ can contribute the value of $w(e)$ twice in $W_{\!\!\text{\tiny\sc Alg}}^{(t^{\mathrm{final}})}$ corresponding to the two variables $x_{e,u}$ and $x_{e,v}$ , the total weight $\Gamma$ of selected edges in our solution is at least

\displaystyle\Gamma\geq W^{\chi+1}+\frac{1}{2}W_{\!\!\text{\tiny\sc Alg}}^{(t^{\mathrm{final}})}\geq\frac{W^{\chi+1}+W_{\!\!\text{\tiny\sc Alg}}^{(t^{\mathrm{final}})}}{2}\geq\frac{W^{\chi+1}+{\mathsf{OPT}}_{\!-W}}{2}=\frac{{\mathsf{OPT}}}{2}\qed

Our proof of Theorem 10.1 is therefore completed once we prove the following lemma.

Lemma 4

For all $i,j\in\{1,\dots,\chi\}$ $\frac{{\widehat{p}}_{i}^{(t^{\mathrm{final}})}}{{\widehat{p}}_{j}^{(t^{\mathrm{final}})}}\leq 4+4\chi$ .

Proof

When $t=t^{\mathrm{final}}$ Case 3 applies and, since the variables $x_{e_{u,j}}\in\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}})}\setminus\widehat{X}_{\mathrm{sol}}^{(t^{\mathrm{final}}-1)}$ are at most $1$ , ${\widehat{q}}_{i}^{(t^{\mathrm{final}})}\leq 0$ and consequently ${\widehat{q}}_{i}^{(t^{\mathrm{final}}-1)}-{\widehat{q}}_{i}^{(t^{\mathrm{final}})}\geq{\widehat{q}}_{i}^{(t^{\mathrm{final}}-1)}$ . Noting that an edge $e\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{u,v\}$ can contribute twice in the various ${\widehat{q}}_{i}^{(t)}$ ’s corresponding to the two variables $x_{e,u}$ and $x_{e,v}$ and remembering that ${\widehat{q}}_{i}^{(0)}={\widehat{q}}_{i}$ , we get

{\widehat{p}}_{i}^{(t^{\mathrm{final}})}\geq\frac{1}{2}\Big{(}\sum_{t=1}^{t^{\mathrm{final}}}\!\!\big{(}{\widehat{q}}_{i}^{(t-1)}-{\widehat{q}}_{i}^{(t)}\big{)}\Big{)}+\mu_{i}\geq\frac{1}{2}\Big{(}\sum_{t=1}^{t^{\mathrm{final}}-1}\!\!\!\big{(}{\widehat{q}}_{i}^{(t-1)}-{\widehat{q}}_{i}^{(t)}\big{)}+{\widehat{q}}_{i}^{(t^{\mathrm{final}}-1)}\Big{)}+\mu_{i}=\frac{{\widehat{q}}_{i}^{(0)}}{2}+\mu_{i}\\ =\frac{{\widehat{q}}_{i}}{2}+\mu_{i}\geq\frac{{\mathsf{OPT}}_{\#}}{2\chi}-\frac{\mu_{i}}{2}+\mu_{i}=\frac{{\mathsf{OPT}}_{\#}}{2\chi}+\frac{\mu_{i}}{2}

We can get an upper bound on ${\widehat{p}}_{i}^{(t^{\mathrm{final}})}$ by getting an upper bound on ${\widehat{q}}_{i}^{(t^{\mathrm{final}}-1)}-{\widehat{q}}_{i}^{(t^{\mathrm{final}})}$ in the following manner. Consider the $\widehat{n}^{(t^{\mathrm{final}})^{\prime}}<\widehat{n}^{(t^{\mathrm{final}})}\leq\chi+1$ nodes $u_{1},\dots,u_{\widehat{n}^{(t^{\mathrm{final}})^{\prime}}}$ in Case 3. By choice of the nodes $v_{1},\dots,v_{\chi+1}$ of degrees $\mathsf{deg}(v_{1}),\dots,\mathsf{deg}(v_{\chi+1})$ , respectively, the number of edges incident on $u_{i}$ is at most $\mathsf{deg}(v_{i})$ for all $i\in\{1,\dots,\widehat{n}^{(t^{\mathrm{final}})^{\prime}}\}$ . Thus, we get ${\widehat{q}}_{i}^{(t^{\mathrm{final}}-1)}-{\widehat{q}}_{i}^{(t^{\mathrm{final}})}\leq\sum_{j=1}^{\chi+1}\mathsf{deg}(v_{j})\leq 2{\mathsf{OPT}}_{\#}$ , and consequently

{\widehat{p}}_{i}^{(t^{\mathrm{final}})}\leq\sum_{t=1}^{t^{\mathrm{final}}}\!\!\big{(}{\widehat{q}}_{i}^{(t-1)}-{\widehat{q}}_{i}^{(t)}\big{)}\leq\sum_{t=1}^{t^{\mathrm{final}}-1}\!\!\big{(}{\widehat{q}}_{i}^{(t-1)}-{\widehat{q}}_{i}^{(t)}\big{)}+2{\mathsf{OPT}}_{\#}={\widehat{q}}_{i}+2{\mathsf{OPT}}_{\#}\\ \leq 2\left({\textstyle\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{i}}\right)+2{\mathsf{OPT}}_{\#}={\textstyle(2+2\chi)\frac{{\mathsf{OPT}}_{\#}}{\chi}}

Thus, for all $i,j\in\{1,\dots,\chi\}$ we have

\frac{{\widehat{p}}_{i}^{(t^{\mathrm{final}})}}{{\widehat{p}}_{j}^{(t^{\mathrm{final}})}}\leq\frac{{(2+2\chi)\frac{{\mathsf{OPT}}_{\#}}{\chi}}}{\frac{{\mathsf{OPT}}_{\#}}{2\chi}+\frac{\mu_{j}}{2}}<4+4\chi\qed

10.1.2 The case of arbitrary $\chi$

As stated below, there are two steps in the previous algorithm that cannot be executed in polynomial time when $\chi$ is not a constant:

(1)

We cannot guess the $\chi+1$ nodes $v_{1},\dots,v_{\chi+1}$ in polynomial time. Instead, we guess only one node $v_{1}$ such that there exists an optimal solution contains $v_{1}$ with the following property: “the remaining $k-1$ nodes in the solution have degree at most $\mathsf{deg}(v_{1})$ ”.

(2)

We cannot guess the exact value of ${\widehat{q}}_{i}$ by exhaustive enumeration and therefore we cannot use the $\chi$ constraints “ $\sum_{u\in\widehat{V}}\sum_{e\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{u,v\}\in C_{i}}x_{e,u}={\widehat{q}}_{i}$ ” in line (3) of the ${\mathsf{LP}}$ -relaxation in Fig. 5 anymore. However, note that it still holds that ${\widehat{q}}_{i}$ is an integer in the set $\big{\{}\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{i},\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{i}+1,\dots,2\big{(}\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{i}\big{)}\big{\}}$ . Thus, instead we use the $2\chi$ constraints

\textbf{(3) }\,\,\,\,\,\,{\textstyle\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{i}}\leq\sum_{u\in\widehat{V}}\sum_{e\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{u,v\}\in C_{i}}x_{e,u}={\widehat{q}}_{i}\leq{\textstyle 2\big{(}\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{i}\big{)}}\,\,\,\,\,\,\text{ for all }i\in\{1,\dots,\chi\}

We need modifications of the bounds in the previous proof to reflect these changes as follows:

$\triangleright$

We make some obvious parameter value adjustments such as: $\widehat{V}=V\setminus\{v_{1}\}$ , $\widehat{E}=E\setminus\{\{u,v_{1}\}\,|\,\{u,v_{1}\}\in E\}$ , $\widehat{k}^{(0)}=k-1$ , $\mu_{i}\leq\mathsf{deg}(v_{1})$ for all $i$ .
$\triangleright$

The number of constraints in lines (1)–(3) of Fig. 5 is now $\widehat{m}+2\chi+1$ .
$\triangleright$

The condition in Case 3 of Fig. 6 is now $1\leq\widehat{n}\leq 2\chi+1$ .
$\triangleright$

In Lemma 2, we select at most $\big{\lfloor}k+\frac{2\chi-1}{2}\big{\rfloor}=k+\chi-1$ nodes.

\triangleright

The calculations for the upper bound for ${\widehat{p}}_{i}^{(t^{\mathrm{final}})}$ in Lemma 4 change as follows. By choice of the node $v_{1}$ of degree $\mathsf{deg}(v_{1})$ , the number of edges incident on $u_{i}$ is at most $\mathsf{deg}(v_{1})$ for all $i\in\{1,\dots,\widehat{n}^{(t^{\mathrm{final}})^{\prime}}\}$ . This now gives ${\widehat{q}}_{i}^{(t^{\mathrm{final}}-1)}-{\widehat{q}}_{i}^{(t^{\mathrm{final}})}\leq(2\chi+1)\mathsf{deg}(v_{1})\leq(2\chi+1){\mathsf{OPT}_{\#}}$ , and therefore ${\widehat{p}}_{i}^{(t^{\mathrm{final}})}\leq{\widehat{q}}_{i}+(2\chi+1){\mathsf{OPT}_{\#}}\leq 2\left({\textstyle\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{i}}\right)+(2\chi+1){\mathsf{OPT}_{\#}}<(2+\chi+2\chi^{2})\frac{{\mathsf{OPT}_{\#}}}{\chi}$ . This gives us the following updated bound:

\frac{{\widehat{p}}_{i}^{(t^{\mathrm{final}})}}{{\widehat{p}}_{j}^{(t^{\mathrm{final}})}}\leq\frac{{(2+\chi+2\chi^{2})\frac{{\mathsf{OPT}}_{\#}}{\chi}}}{\frac{{\mathsf{OPT}}_{\#}}{2\chi}+\frac{\mu_{j}}{2}}<4+2\chi+4\chi^{2}

10.2 The general case: approximating Fmc

Theorem 10.2 (generalizing Theorem 10.1 for Fmc)

We can design a deterministic polynomial-time approximation algorithm for Fmc with the following properties:

(a)

The algorithm selects $\tau$ sets where $\tau\leq\begin{cases}k+\frac{\chi-1}{2},&\mbox{if $\chi=O(1)$}\\ k+\chi-1,&\mbox{otherwise}\end{cases}$
(b)

The algorithm is a $\nicefrac{{1}}{{f}}$ -approximation for Node-fmc, i.e., the total weight of the selected elements is at least $\nicefrac{{{\mathsf{OPT}}}}{{f}}$ .
(c)

The algorithm satisfies the $\varepsilon$ -approximate coloring constraints $($ cf. Inequality (5) $)$ as follows:

for all $i,j\in\{1,\dots,\chi\}$ , $\frac{p_{i}}{p_{j}}<\begin{cases}O(\min\{\chi^{2}f,\,\chi f^{2}\}),&\mbox{if $\chi=O(1)$}\\ O(f^{2}+\chi^{2}f),&\mbox{otherwise}\end{cases}$

The proof of Theorem 10.2 is a suitable modified version of the proof of Theorem 10.1. We point out the important alterations that are needed.

General modifications

$\triangleright$

Nodes and edges now correspond to sets and elements, respectively, incidence of an edge on a node corresponds to membership of an element in a set, and degree of a node correspond to number of elements in a set.
$\triangleright$

There is a set indicator variable $y_{j}$ for every element $\mathcal{S}_{j}\in\widehat{V}$ . For every element $(u_{i},\mathcal{S}_{j})\in\widehat{E}$ , there is an element indicator variable $x_{i,j}$ and a constraint $x_{i,j}=y_{j}$ .
$\triangleright$

Now $\sum_{i=1}^{\chi+1}|\mathcal{S}_{i}|\leq\min\{\chi,f\}{\mathsf{OPT}_{\#}}$ since any element in any one of the sets from $\mathcal{S}_{1},\dots,\mathcal{S}_{\chi+1}$ can appear in at most $\min\{\chi,f\}$ other sets in the collection of sets $\mathcal{S}_{1},\dots,\mathcal{S}_{\chi+1}$ . Also, ${\widehat{q}}_{j}$ is an integer in the set $\Big{\{}\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{j},\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{j}+1,\dots,f\big{(}\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{j}\big{)}\Big{\}}\subseteq\{0,1,2,\dots,n\}$ since any element can appear in at most $f$ sets.
$\triangleright$

An element $u_{i}$ appearing in $f_{i}\leq f$ sets, say sets $\mathcal{S}_{1},\dots,\mathcal{S}_{f_{i}}$ , can now contribute the value of $w(u_{i})$ at most $f_{i}\leq f$ times in $W_{\!\!\text{\tiny\sc Alg}}^{(t^{\mathrm{final}})}$ corresponding to the $f_{i}$ variables $x_{i,1},\dots,x_{i,f_{i}}$ . Thus, we get a $\nicefrac{{1}}{{f}}$ -approximation to the objective function.

Modifications related to $\chi=O(1)$ case

\triangleright

An element $u_{i}$ appearing in $f_{i}\leq f$ sets, say sets $\mathcal{S}_{1},\dots,\mathcal{S}_{f_{i}}$ , can now contribute at most $f_{i}\leq f$ times in the various ${\widehat{q}}_{i}^{(t)}$ ’s corresponding to the $f_{i}$ variables $x_{i,1},\dots,x_{i,f_{i}}$ . This modifies the relevant inequality for ${\widehat{p}}_{i}^{(t^{\mathrm{final}})}$ as follows:

	$\displaystyle{\widehat{p}}_{i}^{(t^{\mathrm{final}})}\geq\frac{{\widehat{q}}_{i}^{(0)}}{f}+\mu_{i}\geq\frac{{\mathsf{OPT}}_{\#}}{f\chi}-\frac{\mu_{i}}{f}+\mu_{i}>\frac{{\mathsf{OPT}}_{\#}}{f\chi}$
	$\displaystyle{\widehat{p}}_{i}^{(t^{\mathrm{final}})}\leq{\widehat{q}}_{i}+\sum_{i=1}^{\chi+1}\|\mathcal{S}_{i}\|\leq{\textstyle f\big{(}\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{i}\big{)}}+\min\{\chi,f\}{\mathsf{OPT}_{\#}}<{\textstyle\min\{\chi^{2}+f,\,2\chi f\}\frac{{\mathsf{OPT}_{\#}}}{\chi}}$
	$\displaystyle\frac{{\widehat{p}}_{i}^{(t^{\mathrm{final}})}}{{\widehat{p}}_{j}^{(t^{\mathrm{final}})}}\leq\frac{{\textstyle\min\{\chi^{2}+f,\,2\chi f\}\frac{{\mathsf{OPT}_{\#}}}{\chi}}}{\frac{{\mathsf{OPT}}_{\#}}{f\chi}}\leq\min\{\chi^{2}f+f^{2},\,2\chi f^{2}\}=O(\min\{\chi^{2}f,\,\chi f^{2}\})$

Modifications related to the arbitrary $\chi$ case

\triangleright

The calculations for the upper bound for ${\widehat{p}}_{i}^{(t^{\mathrm{final}})}$ in Lemma 4 change as follows.

\displaystyle{\textstyle f\big{(}\frac{{\mathsf{OPT}_{\#}}}{\chi}-\mu_{i}\big{)}}+(2\chi+1){\mathsf{OPT}_{\#}}<{\textstyle\big{(}f+\chi+2\chi^{2}\big{)}\frac{{\mathsf{OPT}_{\#}}}{\chi}}<{\textstyle\big{(}f+3\chi^{2}\big{)}\frac{{\mathsf{OPT}_{\#}}}{\chi}}

This gives the final bound of $\frac{{\widehat{p}}_{i}^{(t^{\mathrm{final}})}}{{\widehat{p}}_{j}^{(t^{\mathrm{final}})}}\leq f^{2}+3\chi^{2}f=O(f^{2}+\chi^{2}f)$ .

11 Approximation algorithms for two special cases of Fmc

For approximating these special cases of Fmc, which are still $\mathsf{NP}$ -complete, we will be specific about the various constants and will try to provide approximation algorithms with as tight a constant as we can. For this section, let $\varrho=\max\{\varrho(f),\,\varrho(k)\}$ . Note that $\varrho>1-\nicefrac{{1}}{{{\mathbf{e}}}}$ .

11.1 Segr-fmc: almost optimal deterministic approximation with “at most” $k$ sets

Note that Lemma 1 shows that finding a feasible solution is $\mathsf{NP}$ -complete even for unweighted Segr-fmc with $\chi=2$ . Further inapproximability results for Segr-fmc are stated in Remark 5.

Theorem 11.1

There exists a polynomial-time deterministic algorithm Alg-greed-plus that, given an instance of unweighted Segr-fmc ( $\chi,k$ ) outputs a solution with the following properties:

(a)

The number of selected sets is at most $k$ .

(b)

The approximation ratio is at least $\varrho>1-\nicefrac{{1}}{{{\mathbf{e}}}}$ .

(c)

The coloring constraints are $2$ -approximately satisfied $($ cf. (5) $)$ , i.e.,

\forall\,i,j\in\{1,\dots,\chi\}:\,p_{i}\leq 2\,p_{j}

Remark 5

Based on the $(1-\nicefrac{{1}}{{{\mathbf{e}}}})$ -inapproximability result of Feige in F98 for the maximum $k$ -set coverage problem, it is not difficult to see the two constants in Theorem 11.1, namely $\rho$ and $2$ , cannot be improved beyond $1-\nicefrac{{1}}{{{\mathbf{e}}}}+\varepsilon$ and ${(1-\nicefrac{{1}}{{{\mathbf{e}}}})}^{-1}+\varepsilon\approx 1.58+\varepsilon$ , respectively, for any $\varepsilon>0$ and all $\chi\geq 2$ assuming $\mbox{P}\neq\mathsf{NP}$ .

Remark 6

The “at most $k$ sets” part of the proof arises in the following steps of the algorithm. Since we cannot know $k_{r}$ exactly, we can only assume $\widehat{k_{r}}\leq k_{r}$ since it is possible that the algorithm for the maximum $k$ -set coverage also covers at least $\rho\frac{{\mathsf{OPT}}_{\#}}{\chi}$ elements for some $k<k_{r}$ . Secondly, even if we have the guessed the correct value of $k_{r}$ , the algorithm for the maximum $k_{r}$ -set coverage may cover more than $2\rho\frac{{\mathsf{OPT}}_{\#}}{\chi}$ elements, and thus we have to “un-select” some of the selected sets to get the desired bounds (the proof shows that sometimes we may have to un-select all but one set). The following example shows that a solution that insists on selecting exactly $k$ sets may need to select sets all of which are not in our solution. Consider the following instance of unweighted Fmc( $1,\ell$ ): $\mathcal{U}=\{u_{1},\dotsc,u_{n}\}$ , $\ell=\nicefrac{{n}}{{2}}$ , $\mathcal{S}_{1}=\{u_{1},\dotsc,u_{n/2}\}$ , and $\mathcal{S}_{j+1}=\{u_{(n/2)+j}\}$ for $j=1,\dots,\nicefrac{{n}}{{2}}$ . Our algorithm will select the set $\mathcal{S}_{1}$ whereas any solution that selects exactly $\ell$ sets must selects the sets $\mathcal{S}_{2},\dots,\mathcal{S}_{(n/2)+1}$ .

Proof

We reuse the notations, terminologies and bounds shown in the proof of Theorem 9.1 as needed. Let $\mathcal{U}_{1},\dots,\mathcal{U}_{\chi}$ be the partition of the universe based on the color of the elements, i.e., $\mathcal{U}_{r}=\{u_{\ell}\,|\,\mathcal{C}(u_{\ell})=r\}$ for $r\in\{1,\dots,\chi\}$ . By the definition of Segr-fmc every set contains elements from exactly one such partition and thus, after renaming the sets and elements for notational convenience, we may set assume that our collection $\mathcal{S}_{1},\dots,\mathcal{S}_{m}$ of $m$ sets is partitioned into $\chi$ collection of sets, where the $r^{\mathrm{th}}$ collection (for $r\in\{1,\dots,\chi\}$ ) contains the sets $\mathcal{S}_{1}^{r},\dots,\mathcal{S}_{m_{r}}^{r}$ over the universe $\mathcal{U}_{r}=\{u_{1},\dots,u_{n_{r}}\}$ of $n_{r}$ elements such that $\sum_{r=1}^{\chi}m_{r}=m$ and $\sum_{r=1}^{\chi}n_{r}=n$ . For $r\in\{1,\dots,\chi\}$ and any $\ell$ let Fmc $\!\!{}_{r}(1,\ell)$ be the unweighted Fmc( $1,\ell$ ) problem defined over the universe $\mathcal{U}_{r}$ and the collection of sets $\mathcal{S}_{1}^{r},\dots,\mathcal{S}_{m_{r}}^{r}$ . The following observation holds trivially.

Unweighted Segr-fmc ( $\chi,k$ ) has a valid solution covering $\ell\in\{\chi,2\chi,\dots,\left\lfloor\nicefrac{{n}}{{\chi}}\right\rfloor\chi\}$ elements if and only if (i) for each $r\in\{1,\dots,\chi\}$ , Fmc $\!\!{}_{r}(1,k_{r})$ has a valid solution covering $\nicefrac{{\ell}}{{\chi}}$ elements for some $k_{r}>0$ , and (ii) $\sum_{r=1}^{\chi}k_{r}=k$ .

The above observation suggests that we can guess the value of ${\mathsf{OPT}}_{\#}$ by trying out all possible values of $\ell$ just like the algorithms in Theorem 9.1, and for each such value of $\ell$ we can solve $\chi$ independent Fmc instances and combine them to get a solution of the original Segr-fmc instance. Although we cannot possibly solve the Fmc $\!\!{}_{r}(1,k_{r})$ problems exactly, appropriate approximate solutions of these problems do correspond to a similar approximate solution of Segr-fmc ( $\chi,k$ ) as stated in the following observation:

Suppose that for each $r\in\{1,\dots,\chi\}$ we have a solution $\mathcal{S}_{i_{1}}^{r},\dots,\mathcal{S}_{i_{\widehat{k_{r}}}}^{r}\subseteq\mathcal{U}_{r}$ of Fmc $\!{}_{r}(1,k_{r})$ with the following properties (for some $\eta_{1}\leq 1$ and $\eta_{2}\geq 1$ ): (i) $\eta_{1}(\nicefrac{{\ell}}{{\chi}})\leq|\cup_{p=1}^{\widehat{k_{r}}}\mathcal{S}_{i_{p}}^{r}|\leq\eta_{2}(\nicefrac{{\ell}}{{\chi}})$ , and (ii) $\widehat{k_{r}}\leq k_{r}$ . Then, the collection of sets $\big{\{}\mathcal{S}_{i_{\ell_{r}}}^{r}\,|\,\ell_{r}\in\{1,\dots,\widehat{k_{r}}\},\,r\in\{1,\dots,\chi\}\big{\}}$ outputs a solution of Segr-fmc ( $\chi,k$ ) with the following properties: (a) the number of selected sets is at most $k$ , (b) the number of elements covered is at least $\eta_{1}\ell$ , and (c) for any pair $i,j\in\{1,\dots,\chi\}$ , $\nicefrac{{p_{i}}}{{p_{j}}}\leq\nicefrac{{\eta_{2}}}{{\eta_{1}}}$ .

By the above observation, to prove our claim it suffices if we can find a solution for Fmc $\!{}_{r}(1,k_{r})$ for any $r$ with $\ell={\mathsf{OPT}}_{\#}$ , $\eta_{1}=\varrho$ and $\eta_{2}=2\,\varrho$ . For convenience, we will omit the superscript $r$ from the set labels while dealing with Fmc $\!{}_{r}(1,k_{r})$ . Remove from consideration any sets from $\mathcal{S}_{1},\dots,\mathcal{S}_{m_{r}}$ that contains more than $\nicefrac{{\ell}}{{\chi}}$ elements, and consider the standard (unweighted) maximum $k$ -set coverage problem, that ignores constraint (i) of the above observation, on these remaining collection of sets $\mathcal{T}$ over the universe $\mathcal{U}_{r}$ . Since we have guessed the correct value of $\ell$ , there is at least one valid solution and thus the following assertions hold: (I) there exists a set of $k_{r}$ sets that covers $\frac{{\mathsf{OPT}}_{\#}}{\chi}$ elements, and (II) $|\mathcal{T}|\geq k_{r}$ . Let $\nu_{k}$ denote the maximum number of elements that can be covered by selecting $k$ sets from $\mathcal{T}$ . There are the following two well-known algorithm algorithms for the maximum $k$ -set coverage problem both of which select exactly $k$ sets: the greedy algorithm covers at least $\varrho(k)\nu_{k}$ elements (F98, , Proposition 5.1), where the pipage-rounding algorithm (based on the ${\mathsf{LP}}$ -relaxation in Fig. 2) covers at least $\varrho(f)\nu_{k}$ elements AS04 . Note that we do not know the exact value of $k_{r}$ and we cannot guess by enumerating every possible $k_{r}$ values for every $r\in\{1,\dots,\chi\}$ in polynomial time. To overcome this obstacle, we use the following steps.

$\triangleright$

We run both the algorithms for maximum $k$ -set coverage for $k=1,2,\dots$ until we find the first (smallest) index $\widehat{k_{r}}\leq k_{r}$ such that the better of the two algorithms cover at least $\max\{\varrho(\widehat{k_{r}}),\,\varrho(f)\}\frac{{\mathsf{OPT}}_{\#}}{\chi}\geq\rho\frac{{\mathsf{OPT}}_{\#}}{\chi}$ elements.
$\triangleright$
Suppose that this algorithm selects the $\widehat{k_{r}}$ sets (after possible re-numbering of set indices) $\mathcal{S}_{1},\dots,\mathcal{S}_{\widehat{k_{r}}}$ , where we have ordered the sets such that for every $j\in\{2,3,\dots,\widehat{k_{r}}\}$ the number of elements covered by $\mathcal{S}_{j}$ and not covered by any of the sets $\mathcal{S}_{1},\dots,\mathcal{S}_{j-1}$ is at least as many as the number of elements covered by $\mathcal{S}_{\ell}$ and not covered by any of the sets $\mathcal{S}_{1},\dots,\mathcal{S}_{j-1}$ for any $\ell>j$ . Remember that $\max_{j\in\{1,\dots,\widehat{k_{r}}\}}\{|\mathcal{S}_{j}|\}\leq\nicefrac{{{\mathsf{OPT}}_{\#}}}{{\chi}}$ . Let $j$ be the smallest index such $|\cup_{\ell=1}^{j-1}\mathcal{S}_{j}|<\varrho\frac{{\mathsf{OPT}}_{\#}}{\chi}$ but $|\cup_{\ell=1}^{j}\mathcal{S}_{j}|\geq\varrho\frac{{\mathsf{OPT}}_{\#}}{\chi}$ . We have the following cases.
1. $\triangleright$
  
  If $|\mathcal{S}_{j}|\geq\varrho\frac{{\mathsf{OPT}}_{\#}}{\chi}$ then we select $\mathcal{S}_{j}$ as our solution since $\varrho\frac{{\mathsf{OPT}}_{\#}}{\chi}\leq|\mathcal{S}_{j}|\leq\frac{{\mathsf{OPT}}_{\#}}{\chi}<2\,\varrho\frac{{\mathsf{OPT}}_{\#}}{\chi}$ .
2. $\triangleright$
  
  Otherwise $|\mathcal{S}_{j}|<\varrho\frac{{\mathsf{OPT}}_{\#}}{\chi}$ and in this case we select the $j\leq\widehat{k_{r}}\leq k_{r}$ sets $\mathcal{S}_{1},\dots,\mathcal{S}_{j}$ in our solution since $\varrho\frac{{\mathsf{OPT}}_{\#}}{\chi}\leq|\cup_{\ell=1}^{j}\mathcal{S}_{j}|\leq 2\,\varrho\frac{{\mathsf{OPT}}_{\#}}{\chi}$ .

11.2 $\Delta$ -bal-fmc: improved deterministic approximation

Proposition 2

There exists a polynomial-time deterministic algorithm Alg-greedy that, given an instance of unweighted $\Delta$ -bal-fmc $\!\!(\chi,k)$ outputs a solution with the following properties:

(a): The number of selected sets is $($ exactly $)$ $k$ .
(b): The approximation ratio is at least $\varrho>1-\nicefrac{{1}}{{{\mathbf{e}}}}$ .
(c): The coloring constraints are $O(\Delta f)$ -approximately satisfied $($ cf. (5) $)$ , i.e., $\forall\,i,j\in\{1,\dots,\chi\}:\,\nicefrac{{p_{i}}}{{p_{j}}}\leq(2+2\Delta)f$ .

Proof

As already mentioned in the proof of Theorem 11.1 and elsewhere, there is a deterministic polynomial-time algorithm for the maximum $k$ -set coverage problem with an approximation ratio of $\varrho$ . For the given instance of $\Delta$ -bal-fmc $\!\!(\chi,k)$ , we run this algorithms (ignoring element colors) selecting $k$ sets, say $\mathcal{S}_{1},\dots,\mathcal{S}_{k}$ . Obviously, the total weight of all the elements covered in the selected solution is at least $\varrho\,{\mathsf{OPT}}$ . Let $\alpha^{+}=\sum_{i=1}^{k}\lceil\nicefrac{{|\mathcal{S}_{i}|}}{{\chi}}\rceil+\Delta$ and $\alpha^{-}=\sum_{i=1}^{k}\max\big{\{}1,\,\lfloor\nicefrac{{|\mathcal{S}_{i}|}}{{\chi}}\rfloor-\Delta\big{\}}$ . Note that $k\leq\alpha^{-}\leq\alpha^{+}\leq\alpha^{-}+(2\Delta+1)k$ . Since each of the sets in the solution is balanced, an upper bound for the number $p_{i}$ of elements of color $i$ in the solution is given by $p_{i}\leq\alpha^{+}$ . Also note that by definition of $f$ we have $p_{i}\geq\frac{\alpha^{-}}{f}$ . It thus follows that for any $i$ and $j$ we have $\nicefrac{{p_{i}}}{{p_{j}}}\leq f\times\frac{\alpha^{+}}{\alpha^{-}}\leq(2+2\Delta)f$ .

12 Approximating Geom-fmc via randomized shifting

We refer the reader to textbooks such as V01 for a general overview of the randomized shifting technique (textbook V01 illustrates the technique in the context of Euclidean travelling salesperson problem).

Theorem 12.1

For any constant $0<\varepsilon<1$ , we can design a randomized algorithm Alg-geom for Geom-fmc with the following properties:

(a)

Alg-geom runs in $O((\Delta/d)^{d}2^{(Cd/\varepsilon)^{O(d)}}k^{d})$ time.
(b)
Alg-geom satisfies the following properties with probability $1-o(1)$ $($ cf. Inequality (5)^′–(5) ${}^{\prime\prime})$ :
1. $\triangleright$
  
  The algorithm covers at least $(1-O(\varepsilon))({\mathsf{OPT}}-\varepsilon\chi)$ points.
2. $\triangleright$
  
  The algorithm satisfies the $(1+\varepsilon)$ -approximate coloring constraints $($ cf. Inequality (5) ${}^{\prime\prime})$ , i.e., for all $i,j\in\{1,\dots,\chi\}$ , $p_{i}\leq(1+\varepsilon)p_{j}$ .

Remark 7

In many geometric applications, the dimension parameter $d$ is a fixed constant. For this case, Alg-geom runs in polynomial time, and moreover, under the mild assumption of $\frac{{\mathsf{OPT}}}{\chi}\geq\eta$ for some constant $\eta>1$ , Alg-geom covers at least $(1-O(\varepsilon)){\mathsf{OPT}}$ points, i.e., under these conditions Alg-geom behaves like a randomized polynomial-time approximation scheme.

Proof

Fix an optimal solution having $k$ unit balls $\mathcal{B}_{1}^{\ast},\ldots,\mathcal{B}_{k}^{\ast}\subset{\mathbb{R}}^{d}$ , such that for all $i,j\in\{1,\dots,\chi\}$ , $\mu_{i}(\mathcal{B}^{\ast})=\mu_{j}(\mathcal{B}^{\ast})$ , where $\mathcal{B}^{\ast}=\bigcup_{i=1}^{k}\mathcal{B}_{i}^{\ast}$ . Thus, we need to show that our algorithm Alg-geom computes in $(\Delta/d)^{d}2^{(Cd/\varepsilon)^{O(d)}}k^{d}$ time a set of unit balls $B_{1},\ldots,B_{k}\subset\mathbb{R}^{d}$ such that the following assertions hold with probability $1-o(1)$ (where $B=\bigcup_{i=1}^{k}B_{i}$ ):

	$\displaystyle\sum_{i=1}^{\chi}\mu_{i}(B)>(1-O(\varepsilon))\sum_{i=1}^{\chi}(\mu_{i}(\mathcal{B}^{\ast})-\varepsilon)$
	$\displaystyle\forall\,i,j\in\{1,\ldots,\chi\}:\,\mu_{i}(B)\leq(1+\varepsilon)\mu_{j}(B)$

Set $L=8d/\varepsilon$ . Let $G\subset\mathbb{R}^{d}$ be an axis-parallel grid such that every connected component of ${\mathbb{R}}^{d}\setminus G$ is an open $d$ -dimensional hypercube isometric to $(0,L)^{d}$ . In other words, $G$ is the union of $d$ infinite families of axis-parallel $(d-1)$ -dimensional hyperplanes, spaced apart by $L$ in each orthonormal direction. Let $\alpha\in[0,L)$ be chosen uniformly at random, and let $G^{\prime}=G+\alpha$ be the random translation of $G$ by $\alpha$ , i.e.,

(p_{1}+\alpha,\dots,p_{d}+\alpha)\in G^{\prime}\,\,\equiv\,\,(p_{1},\dots,p_{d})\in G

Let $F$ is the set of indices of all balls $\mathcal{B}_{i}^{\ast}$ that has a non-empty intersection with the randomly shifted grid $G^{\prime}$ , i.e.,

F=\{i\in\{1,\dots,k\}:\mathcal{B}^{\ast}_{i}\cap G^{\prime}\neq\emptyset\}

Let $\mathcal{B}^{\ast,F}=\bigcup_{i\in F}\mathcal{B}_{i}^{\ast}$ . Any point $p\in{\mathbb{R}}^{d}$ is contained in $\mathcal{B}^{\ast,F}$ only if it is contained in some unit ball intersecting $G^{\prime}$ . Therefore, $p\in\mathcal{B}^{\ast,F}$ only if it is at distance at most $2$ from $G^{\prime}$ (in other words, $\mathcal{B}^{\ast,F}$ is contained in the $2$ -neighborhood of $G^{\prime}$ ). The probability that any particular point $p$ is at distance at most $2$ from any family of parallel randomly shifted hyperplanes in $G^{\prime}$ is exactly $4/L$ . By the union bound over all dimensions, $\Pr[p\in\mathcal{B}^{\ast,F}]\leq 4d/L$ . Therefore, by the linearity of expectation,

{\mathbb{E}}[\mu_{i}(\mathcal{B}^{\ast,F})]=\sum_{p\in\cup_{i=1}^{k}\mathcal{B}_{i}^{\ast}}\!\!\!\!\Pr\left[p\in\mathcal{B}^{\ast,F}\,\right]\,\mu_{i}(p)\leq\frac{4d}{L}\mu_{i}(\mathcal{B}^{\ast})

Consequently, by Markov’s inequality, we get

\Pr\left[\textstyle\mu_{i}(\mathcal{B}^{\ast,F})\geq\frac{8d}{R}\mu_{i}(\mathcal{B}^{\ast})\right]\leq\nicefrac{{1}}{{2}}

Set $\delta=2^{-\Theta(d)}\varepsilon/C$ . Let $B_{1}^{\ast\ast},\ldots,B_{k}^{\ast\ast}$ be the collection of unit balls in ${\mathbb{R}}^{d}$ obtained as follows. For each $i\in\{1,\dots,k\}\setminus F$ , obtain a unit ball by translating $\mathcal{B}_{i}^{\ast}$ such that its center has coordinates that are integer multiples of $\delta$ , i.e., it is an element of the dilated integer lattice $\delta\cdot\mathbb{Z}^{d}$ . For every $i\in F$ , we obtain a unit ball by picking an arbitrary ball obtained for some $j\in\{1,\dots,k\}\setminus F$ as described above. Essentially, the new solution $B^{\ast\ast}_{1},\dots,B^{\ast\ast}_{k}$ is missing all the balls that intersect $G^{\prime}$ , and rounds every other ball so that its center is contained in some integer lattice. In this construction, each ball $\mathcal{B}_{i}^{\ast}$ , with $i\notin F$ , gets translated by at most some distance $\sqrt{d}\delta$ . Since each for all $j\in\{1,\ldots,\chi\}$ , $\mu_{j}$ is $C$ -Lipschitz, it follows that

|\mu_{j}(\mathcal{B}^{\ast}_{i})-\mu_{j}(B^{\ast\ast}_{i})|\leq\mathrm{vol}(\mathcal{B}^{\ast}_{i})\,\sqrt{d}\,\delta\,C\leq 2^{\Theta(d)}\delta C

Letting $B^{\ast\ast}=\bigcup_{i=1}^{k}B^{\ast\ast}_{i}$ , we get that for all $i\in\{1,\dots,\chi\}$ ,

|\mu_{j}(\mathcal{B}^{\ast})-\mu_{j}(B^{\ast\ast})|\leq k2^{\Theta(d)}\delta C

Let ${\cal I}$ be the set of connected components of $[0,\Delta]^{d}\setminus G^{\prime}$ . We refer to the elements of ${\cal I}$ as cells. For each $A\in{\cal I}$ , we enumerate the set, ${\cal S}_{A}$ , of all possible subsets of at most $k$ unit balls with centers in $A\cap(\delta\cdot\mathbb{Z}^{d})$ . There are at most $(L/\delta)^{d}$ lattice points in $A$ , and thus there are at most $2^{(L/\delta)^{d}}$ such subsets of unit balls. Since $|{\cal I}|\leq(\lceil\Delta/L\rceil)^{d}$ , it follows that this enumeration takes $O((\Delta/L)^{d}\,2^{(L/\delta)^{d}})$ time.

For each enumerated subset $J\in{\cal S}_{A}$ of unit balls, we record the vector

\left(|J|,\frac{\varepsilon}{k}\left\lfloor\mu_{1}(X)\frac{k}{\varepsilon}\right\rfloor,\dots,\frac{\varepsilon}{k}\left\lfloor\mu_{k}(X)\frac{k}{\varepsilon}\right\rfloor\right)

where $X=\bigcup_{Y\in J}Y$ . There are at most $(2^{O(d)}k/\varepsilon)^{d}$ such vectors for each cell in ${\cal I}$ . Via standard dynamic programming, we can inductively compute all possible sums of vectors such that we pick at most one vector from each cell, and the total sum of the first coordinate, i.e., the number of unit balls, is at most $k$ . This can be done in $O((\Delta/L)^{d}2^{(L/\delta)^{d}}(2^{O(d)}k/\varepsilon)^{d})$ time. For the correct choice of vectors that corresponds to the solution $B^{\ast\ast}$ , the sum of the vectors we compute is correct up to an additive factor of $\varepsilon$ on each coordinate. This means that we compute a solution $B_{1},\ldots,B_{k}$ , with the following property:

\sum_{i=1}^{k}\mu_{i}(B)\geq(1-\varepsilon)\sum_{i=1}^{\chi}\mu_{i}(B^{\ast\ast})\geq(1-\varepsilon)\Big{[}\sum_{i=1}^{\chi}\big{(}\,\mu_{i}(\mathcal{B}^{\ast})-2^{\Theta(d)}\delta C-\mu_{i}(\mathcal{B}^{\ast,F})\,\big{)}\Big{]}\\ \geq\big{(}\,1-\varepsilon-(8d/L)\,\big{)}\Big{[}\sum_{i=1}^{\chi}\big{(}\,\mu_{i}(\mathcal{B}^{\ast})-2^{\Theta(d)}\delta C\,\big{)}\Big{]}=\big{(}\,1-2\varepsilon\,\big{)}\Big{[}\sum_{i=1}^{\chi}\big{(}\,\mu_{i}(\mathcal{B}^{\ast})-\varepsilon\,\big{)}\Big{]}

with probability at least $\nicefrac{{1}}{{2}}$ . Repeating the algorithm $O(\log n)$ times and returning the best solution found, results in the high-probability assertion, which concludes the proof.

13 Conclusion and open problems

In this paper we formulated a natural combinatorial optimization framework for incorporating fairness issues in coverage problems and provided a set of approximation algorithms for the general version of the problem as well as its special cases. Of course, it is possible to design other optimization frameworks depending on the particular application in hand, and we encourage researchers to do that. Below we list some future research questions related to our framework:

Eliminating the gap of factor $f$ in ${\mathsf{LP}}$ -relaxation:: As noted in Section 9.7, all of our ${\mathsf{LP}}$ -relaxations incur a gap of factor $f$ in the coloring constraints while rounding. It seems non-trivial to close the gap using additional linear inequalities while preserving the same approximation ratio. However, it may be possible to improve the gap using ${\mathsf{SDP}}$ -relaxations.
Primal-dual schema:: Another line of attack for the Fmc problems is via the primal-dual approach V01 . For example, can the primal-dual approach for partial coverage problem by Gandhi, Khuller and Srinivasan GKS04 be extended to Fmc? A key technical obstacle seems to center around effective interpretation of the dual of the coloring constraints. Our iterated rounding approach was able to go around this obstacle but the case when $\chi=\omega(1)$ may be improvable.
Fixed parameter tractability:: As mentioned in Section 4 fixed-parameter tractability issues for $k$ -node coverage have been investigated by prior researchers such as Marx M08 and Gupta, Lee and Li in GLL18a ; GLL18b . It would be interesting to extend these results to Node-fmc.
Generalizing to non-decreasing submodular set objective functions:: The proofs and proof techniques in this paper do not generalize to the case when the objective function for our Fmc problems is a (more general) non-decreasing submodular set function. It would be interesting to devise new algorithmic techniques and proofs for this more general case. Approximation algorithms for such generalizations for the standard maximum $k$ -set coverage problem (see Section 4) were provided in KST09 .

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Appendix

Proof of Lemma 1

(a) We describe the proof for $\chi=2$ ; generalization to $\chi>2$ is obvious. The reduction is from the Exact Cover by 3-sets (X3C) problem which is defined as follows. We are given an universe $\mathcal{U}^{\prime}=\{u_{1},\dots,u_{n^{\prime}}\}$ of $n^{\prime}$ elements for some $n^{\prime}$ that is a multiple of $3$ , and a collection of $n^{\prime}$ subsets $\mathcal{S}_{1},\dots,\mathcal{S}_{n^{\prime}}$ of $\mathcal{U}$ such that $\bigcup_{j=1}^{n^{\prime}}\mathcal{S}_{j}=\mathcal{U}$ , every element of $\mathcal{U}^{\prime}$ occurs in exactly $3$ sets and $|\mathcal{S}_{j}|=3$ for $j=1,\dots,n^{\prime}$ . The goal is to decide if there exists a collection of $\nicefrac{{n^{\prime}}}{{3}}$ (disjoint) sets whose union is $\mathcal{U}^{\prime}$ . X3C is known to be $\mathsf{NP}$ -complete GJ79 . Given an instance $\langle\mathcal{U}^{\prime},\mathcal{S}_{1},\dots,\mathcal{S}_{n^{\prime}}\rangle$ of X3C as described, we create the following instance $\langle\mathcal{U},\mathcal{S}_{1},\dots,\mathcal{S}_{n^{\prime}+1},k\rangle$ of Fmc( $2,k$ ):

(i)

The universe is $\mathcal{U}=\{u_{1},\dots,u_{n^{\prime}}\}\cup\{u_{n^{\prime}+1},\dots,u_{2n^{\prime}}\}$ (and thus $n=2n^{\prime}$ ),
(ii)

$w(u_{j})=1$ for $j=1,\dots,2n^{\prime}$ ,
(iii)

the sets are $\mathcal{S}_{1},\dots,\mathcal{S}_{n^{\prime}}$ and a new set $\mathcal{S}_{n^{\prime}+1}=\{u_{n^{\prime}+1},\dots,u_{2n^{\prime}}\}$ ,
(iv)

the coloring function is given by $\mathcal{C}(u_{j})=\left\{\begin{array}[]{r l}1,&\mbox{if $1\leq j\leq n^{\prime}$}\\ 2,&\mbox{otherwise}\end{array}\right.$ , and
(v)

$k=\frac{n^{\prime}}{3}+1=\frac{n}{6}+1$ .

Clearly, every element of $\mathcal{U}$ occurs in no more than $3$ sets and all but the set $\mathcal{S}_{n^{\prime}+1}$ contains exactly $3$ elements. The proof is completed once the following is shown:

( $\ast$ ) the given instance of X3C has a solution if and only if the transformed instance of Fmc( $2,1+\nicefrac{{n}}{{6}}$ ) has a solution.

A proof of ( $\ast$ ) is easy: since the set $\mathcal{S}_{n^{\prime}+1}$ must appear in any valid solution of Fmc, a solution $\mathcal{S}_{i_{1}},\dots,\mathcal{S}_{i_{n^{\prime}/3}}$ of X3C corresponds to a solution $\mathcal{S}_{i_{1}},\dots,\mathcal{S}_{i_{n^{\prime}/3}},\mathcal{S}_{n^{\prime}+1}$ of Fmc( $2,k$ ) and vice versa.

(b) The proof is similar to that in (a) but now instead of X3C we reduce the node cover problem for cubic (i.e., $3$ -regular) graphs (VC₃) which is defined as follows: given a cubic graph $G=(V,E)$ of $n^{\prime}$ nodes and $3n^{\prime}/2$ edges and an integer $k^{\prime}$ , determine if there is a set of $k^{\prime}$ nodes that cover all the edges. $VC_{3}$ is known to be $\mathsf{NP}$ -complete even if $G$ is planar GJ79 . For the translation to an instance of Fmc( $2,k$ ), edges of $G$ are colored with color $1$ , we add a new connected component $\mathcal{K}_{(3n^{\prime}/2)+1}$ to $G$ that is a complete graph of $(3n^{\prime}/2)+1$ nodes with every edge having color $2$ , transform this to the set-theoretic version of Fmc using the standard transformation from node cover to set cover and set $k=k^{\prime}+1$ ; note that $n=3n^{\prime}/2+\binom{(3n^{\prime}/2)+1}{2}=\Theta((n^{\prime})^{2})$ and $a=3n^{\prime}/2=O(\sqrt{n}\,)$ . To complete the proof, note that any feasible solution for the Fmc( $2,k$ ) instance must contain exactly one node from $\mathcal{K}_{(3n^{\prime}/2)+1}$ covering $3n^{\prime}/2$ edges and therefore the solution for the edges with color $1$ must correspond to a node cover in $G$ (and vice versa).

(c) We given a different reduction from X3C. Given an instance $\langle\mathcal{U}^{\prime},\mathcal{S}_{1},\dots,\mathcal{S}_{n^{\prime}}\rangle$ of X3C as in (a), we create the following instance $\langle\mathcal{U},\mathcal{T}_{1},\dots,\mathcal{T}_{n^{\prime}},k\rangle$ of Fmc( $n^{\prime},k$ ):

(i)

For every set $\mathcal{S}_{i}=\big{\{}u_{i_{1}},u_{i_{2}},u_{i_{3}}\big{\}}$ of X3C we have three elements $u_{i_{1}}^{i},u_{i_{2}}^{i},u_{i_{3}}^{i}$ and a set $\mathcal{T}_{i}=\big{\{}u_{i_{1}}^{i},u_{i_{2}}^{i},u_{i_{3}}^{i}\big{\}}$ in Fmc (and thus $n=3n^{\prime}$ , $a=3$ and $f=1$ ),
(ii)

$w(u_{i_{j}}^{i})=1$ and $\mathcal{C}(u_{i_{j}}^{i})=i_{j}$ for $i\in\{1,\dots,n^{\prime}\},j\in\{1,2,3\}$ (and thus $\chi=n^{\prime}=\nicefrac{{n}}{{3}}$ ),
(iii)

$k=\nicefrac{{n^{\prime}}}{{3}}=\nicefrac{{n}}{{9}}$ .

The proof is completed by showing the given instance of X3C has a solution if and only if the transformed instance of Fmc( $\nicefrac{{n}}{{3}},\nicefrac{{n}}{{9}}$ ) has a solution. This can be shown as follows. We include the set $\mathcal{T}_{i}$ in the solution for Fmc if and only if the set $\mathcal{S}_{i}$ is in the solution for X3C. For any valid solution of X3C and every $j\in\{1,\dotsc,n^{\prime}\}$ the element $u_{j}\in\mathcal{U}^{\prime}$ appears in exactly one set, say $\mathcal{S}_{\ell}=\big{\{}u_{\ell_{1}},u_{\ell_{2}},u_{\ell_{3}}\big{\}}$ , of X3C where one of the elements, say $u_{\ell_{1}}$ , is $u_{j}$ . Then, the solution of Fmc contains exactly one element, namely the element $u_{\ell_{1}}^{\ell}$ , of color $\ell_{1}=j$ . Conversely, given a feasible solution of Fmc with at most $k\leq\nicefrac{{n}}{{9}}$ sets, first note that if $k<\nicefrac{{n}}{{9}}$ then the total number of colors of various elements in the solution is $3k<n^{\prime}$ and thus the given solution is not valid. Thus, $k=\nicefrac{{n}}{{9}}$ and therefore the solution of X3C contains $\nicefrac{{n}}{{9}}=\nicefrac{{n^{\prime}}}{{3}}$ sets. Now, for every color $j$ the solution of Fmc contains a set, say $\mathcal{T}_{\ell}=\big{\{}u_{\ell_{1}}^{\ell},u_{\ell_{2}}^{\ell},u_{\ell_{3}}^{\ell}\big{\}}$ , containing an element of color $j$ , say the element $u_{\ell_{1}}^{\ell}$ . Then $\ell_{1}=j$ and the element $u_{j}$ appears in a set in the solution of X3C. To see that remaining claims about the reduction, there is no solution of Fmc that includes at least one element of every color and that is not a solution of X3C.

Maximizing coverage while ensuring fairness: a tale of conflicting objectives

Abstract

Keywords:

1 Introduction

1.1 Different research perspectives in ensuring fairness

2 Fair maximum coverage: notations, definitions and related concepts

2.1 Three special cases of the general version of Fmc

3 Sketch of application scenarios

4 Review of prior related works

5 Summary of our contribution

5.1 Feasibility hardness results

5.2 Algorithmic results

5.3 Remarks on proof techniques

6 Organization of the paper and proof structures

7 Computational hardness of finding a feasible solution of Fmc

Lemma 1

Remark 1

8 Relaxing coloring constraints for algorithmic designs

9 𝖫𝖯{\mathsf{LP}}-relaxation based randomized approximation algorithms for Fmc

Theorem 9.1

Remark 2

Remark 3

Remark 4

9.1 An obvious generalization of 𝖫𝖯{\mathsf{LP}}-relaxation of maximum kk-set coverage fails

9.2 Alg-large-opt#: strengthening 𝖫𝖯{\mathsf{LP}}-relaxation via additional inequalities

9.3 A general technique for obtaining joint high-probability statement

9.4 Alg-large-opt#: further details and relevant analysis

Fact 1

9.5 Alg-medium-opt#: details and proofs of relevant claims in (a)–(c) and (d)(ii)

9.6 Alg-small-opt#: details and proofs of relevant claims in (a)–(c) and (d)(iii)

9.7 Limitations of our 𝖫𝖯{\mathsf{LP}}-relaxation: “a gap of factor ff” for coloring constraints

Proposition 1

Proof

10 A tale of fewer colors: deterministic approximation for Fmc when χ\chi is “not too large”

Fact 2 (Rank lemma)

10.1 Approximating Node-fmc

Theorem 10.1

10.1.1 The case of χ=O​(1)\chi=O(1)

Lemma 2

Proof

Lemma 3

Proof

Lemma 4

Proof

10.1.2 The case of arbitrary χ\chi

10.2 The general case: approximating Fmc

Theorem 10.2 (generalizing Theorem 10.1 for Fmc)

General modifications

Modifications related to χ=O​(1)\chi=O(1) case

Modifications related to the arbitrary χ\chi case

11 Approximation algorithms for two special cases of Fmc

11.1 Segr-fmc: almost optimal deterministic approximation with “at most” kk sets

Theorem 11.1

Remark 5

Remark 6

Proof

11.2 Δ\Delta-bal-fmc: improved deterministic approximation

Proposition 2

Proof

12 Approximating Geom-fmc via randomized shifting

Theorem 12.1

Remark 7

Proof

13 Conclusion and open problems

References

Proof of Lemma 1

9 ${\mathsf{LP}}$ -relaxation based randomized approximation algorithms for Fmc

9.1 An obvious generalization of ${\mathsf{LP}}$ -relaxation of maximum $k$ -set coverage fails

9.2 Alg-large-opt_#: strengthening ${\mathsf{LP}}$ -relaxation via additional inequalities

9.4 Alg-large-opt_#: further details and relevant analysis

9.5 Alg-medium-opt_#: details and proofs of relevant claims in (a)–(c) and (d)(ii)

9.6 Alg-small-opt_#: details and proofs of relevant claims in (a)–(c) and (d)(iii)

9.7 Limitations of our ${\mathsf{LP}}$ -relaxation: “a gap of factor $f$ ” for coloring constraints

10 A tale of fewer colors: deterministic approximation for Fmc when $\chi$ is “not too large”

10.1.1 The case of $\chi=O(1)$

10.1.2 The case of arbitrary $\chi$

Modifications related to $\chi=O(1)$ case

Modifications related to the arbitrary $\chi$ case

11.1 Segr-fmc: almost optimal deterministic approximation with “at most” $k$ sets

11.2 $\Delta$ -bal-fmc: improved deterministic approximation