On the Power of Static Assignment Policies for Robust Facility Location Problems

Let us begin with a formal problem definition. We are given a set of $n$ facilities $\cal F$ and $m$ clients $\cal C$ in a common metric $d$ where $d_{ij}$ denotes the distance between $i$ and $j$. For each facility $i\in{\cal F}$, there is a cost $c_{i}$ per unit of supply at $i$. The demand uncertainty is modeled by an implicit set of scenarios ${\cal C}^{k}$ that includes all subsets of clients $\cal C$ of size at most $k$. The decision-maker needs to select an (integral) number of units of supply $x_{i}$ for each facility $i\in\cal F$ in the first-stage. An adversary observes the first stage decisions and selects a worst-case demand scenario $S\in{\cal C}^{k}$ that must be satisfied with the first-stage supply, where each client in the realized scenario needs one unit of supply. The goal is to minimize the sum of the first-stage supply cost and the worst-case assignment cost over all second-stage demand scenarios. We refer to this problem as soft-capacitated robust facility location (SCRFL). Typically in the literature, soft-capacitated refer to settings where violations of capacity upper bounds are allowed. The analogue here is that we can add any amount of supply in a facility without upper bounds ($x_{i}\in\mathbb{Z_{+}})$ but we pay a per unit cost of supply.

We consider a class of static assignment policies, where each of the $m$ clients has a static fractional assignment to facilities that is independent of the scenario, leading to a feasible second-stage solution for each demand scenario, while respecting supply capacities. Note this is a restriction, since the optimal second-stage assignment decisions are scenario-dependent in general. As a warm-up, we show that static assignment policies are optimal for the uncapacitated case with unlimited supply at each open facility (i.e., there is a cost $c_{i}$ to open facility $i$ with unlimited supply). We refer to this problem as uncapacitated robust facility location (URFL). This is based on the intuition that each client can be assigned to the closest open facilities in an optimal solution in any scenario; this leads to optimality of a static assignment policy for the LP relaxation (Theorem 1.1).

The optimality of static assignment is not true in general when the supply at facilities is constrained (or equivalently, there is a cost per unit of supply). The main contribution in this paper is to show that static assignment policies give an $O(\log k/\log\log k)$-approximation for the LP relaxation of (SCRFL) (Theorem 1.2). We show this by constructing such a solution, starting from an optimal first-stage supply. The optimal static assignment policies can be computed efficiently by solving a compact LP.

Furthermore, the fractional supply in the first stage can be rounded to an integral supply using ideas similar to rounding algorithms for the deterministic facility location [18]. In particular, the static assignment solution for the uncapacitated case can be rounded to give a $4$-approximation algorithm for (URFL). The static assignment solution for the soft-capacitated case can be rounded within a constant factor, which results in an $O(\log k/\log\log k)$-approximation algorithm for (SCRFL). We would like to note that while the fractional assignment is static in our approximate LP solution, our integral assignment for any client in the second-stage depends on the other demand clients in the scenario; thereby, making our static assignment policy adaptive in implementation.

In this section, we consider the uncapacitated robust facility location problem (URFL) where for each $i\in{\cal F}$, there is a cost $c_{i}$ to open facility $i$ with unlimited supply. The problem can be stated as the following integer program, where each binary variable $x_{i},i\in{\cal F}$ indicates if facility $i$ is opened and each $y_{ij}^{S},i\in{\cal F},j\in S,S\in{\cal C}^{k}$ indicates the assignment of client $j$ to facility $i$ in scenario $S$.

Note that the second-stage problem is a transportation problem and since the demand of a client is integral (0 or 1), the optimal solution $y^{S}_{ij}$ is integral as well. The special case of (URFL) where the uncertainty set contains only a single scenario corresponds to the NP-hard classical and well-studied uncapacitated facility location problem, which is hard to approximate within a constant better than 1.463 unless NP has an $O(n^{O(\log\log n)})$-time algorithm [8]. We let (LP-URFL) denote the linear relaxation of (URFL), where we replace $x_{i}\in\{0,1\}$ by $x_{i}\geq 0$ for each $i\in{\cal F}$. We would like to note that, in our model, the number of scenarios could be exponential in the dimension of the problem. Hence, in general, even the linear relaxation of such a problem could be challenging. However, we show that (LP-URFL) can be solved in polynomial time using a Static Assignment Policy for the second-stage variables. Moreover, we can round the fractional solution losing only a constant factor, thereby getting a constant approximation for (URFL). This section serves as a warm-up for introducing and motivating static assignment policies before addressing the class of capacitated robust facility location problems.

Consider an optimal solution of (URFL). Since each open facility can have an unlimited amount of supply, each client in the realized scenario is assigned to the closest facility among the opened ones. Therefore, a client $j$ is always assigned to the same open facility in all scenarios $S$ where $j\in S$. The same observation holds as well for (LP-URFL) where each client is assigned to the same fractionally opened facilities independent of the realized scenario. Thus, the assignment of a client is static. This can be captured by the following policy.

Let (Static-URFL) denote the problem after restricting the second-stage variables $y_{ij}^{S}$ in (LP-URFL) to a policy (1), which can then be reformulated as follows:

From Theorem 1.1, (Static-URFL) is equivalent to (LP-URFL). The number of variables in (Static-URFL) is reduced to a polynomial number since the $y_{ij}$ no longer depend on the scenario $S$. The inner maximization problem is still taken over an exponential number of scenarios; however we can separate efficiently over these scenarios and write an efficient compact LP formulation for (Static-URFL):

where the first equality holds because the optimal solution of the right maximization problem occurs at the extreme points of the $k$-ones polytope, which corresponds to the worst-case scenarios of ${\cal C}^{k}$ and the second equality follows from strong duality. Therefore, by dropping the min and introducing $\mu$ and all $\omega_{j}$ as variables, we reformulate (Static-URFL) as the following linear program:

Finally, we round the solution of (Static-URFL) to an integral solution for (URFL) while losing only a constant factor. This can be done using prior work on rounding techniques from the literature of deterministic facility location problems. In fact, the LP rounding technique in Shmoys et al. [18], which gives a $4$-approximation algorithm to the deterministic uncapacitated problem also gives a $4$-approximation algorithm for (Static-URFL). The idea is to define a ball around each client of radius equal to the fractional assignment cost of the client (which is independent of any scenario for our static policy). Then, we open facilities in non-intersecting balls of ascending radius. The result is given in the following theorem and for completeness, the details of the rounding are in Appendix 0.A.

We would like to note that the focus in this section is not about finding the best constant approximation for (URFL), but we introduce it as a warm-up for motivating the static assignment policy before presenting our main result in the next section.

In this section, we consider the soft-capacitated robust facility location (SCRFL) which is similar to (URFL) except that each facility $i$ incurs a linear supply cost, where $c_{i}$ is the cost per unit of supply. We refer to $x_{i}$ as the supply (or capacity) in facility $i$. Each client in the realized scenario needs to be satisfied by one unit of supply. The problem is called soft-capacitated since there is no upper bound on $x_{i}$, ($x_{i}\in\mathbb{Z_{+}})$. The problem is given by the following integer program:

We let (LP-SCRFL) denote the linear relaxation of (SCRFL), where we replace $x_{i}\in\mathbb{N}$ by $x_{i}\geq 0$, for each $i\in{\cal F}$. We would like to note that even the linear relaxation (LP-SCRFL) is challenging to solve since it has exponentially-many variables (scenarios). Unlike the uncapacitated case, the static assignment policy (1) is not optimal for (LP-SCRFL) and the optimal assignment for each client depends, in general, on the realized scenario. In particular, the same client could be assigned to different facilities in different scenarios. In contrast, we show the surprising result that a static assignment policy gives $O(\log k/\log\log k)$-approximation to (LP-SCRFL). Moreover, we can round the solution of the static assignment policy to an integral solution for (SCRFL) and only lose an additional constant factor. We let (Static-SCRFL) denote the problem when we restrict the second-stage variables $y_{ij}^{S}$ in (LP-SCRFL) to static assignment policies (1). The problem can then be reformulated as follows:

Our main contribution in this section is to show that a static assignment policy (1) gives $O(\log k/\log\log k)$-approximation for (LP-SCRFL) (Theorem 1.2). To prove this theorem, we consider an optimal solution of (LP-SCRFL) and massage it to construct a solution of the form (1) while losing $O(\log k/\log\log k)$ factor. We first present our construction and several structural lemmas and then give the proof of Theorem 1.2.

In the procedure, while the set $C$ is not empty, we pick a client $j\in C$ and grow three balls around it: $B_{j}^{2\ell-1}$ (internal ball), $B_{j}^{2\ell}$ (medium ball) and $B_{j}^{2\ell+1}$ (external ball) starting with $\ell=1$. For each $\ell$, we check if the number of clients in the internal ball $B_{j}^{2\ell-1}$ is greater than $k$ (line 4); if this is the case, we remove them from $C$, put them in $C_{1}$ and restart in line 2. If not, we check if the supply in the medium ball $B_{j}^{2\ell}$ is sufficient to satisfy half of the clients in the internal ball $B_{j}^{2\ell-1}$ (line 7); if that is not the case, we remove those clients from $C$, put them in $C_{2}$, and restart in line 2. Otherwise, we finally check if the supply in the the medium ball $B_{j}^{2\ell}$ is sufficient to satisfy a fraction $1/2\alpha$ of the clients in the external ball $B_{j}^{2\ell+1}$ (line 10); if that is the case we remove all the clients in $B_{j}^{2\ell+1}$ and put them in $C_{3}$, we also remove all the facilities in $B_{j}^{2\ell}$ and restart in line 2. If none of these three conditions holds, we increase $\ell$ to $\ell+1$. First, we show that after at most $\log_{\alpha}k$ increments (i.e., $\ell\leq\log_{\alpha}k$), one of three conditions must hold and therefore we will remove some clients from $C$ and restart in line 2. Which implies that after a finite number of iterations, the set $C$ becomes empty. In particular, We have the following lemma.

Now we are ready to present our static assignment policy for (LP-SCRFL). The following three lemmas show our constructed static assignment for each client in the three subsets $C_{1},C_{2},C_{3}$. Moreover, we specify the supply used to satisfy each subset of these clients and present the analysis for the assignment cost.

For each client in the set $C_{1}$, we know that it belongs to a ball with at least $k$ clients. By the feasibility of the optimal solution, this implies that there exists $k$ units of supply in $\boldsymbol{x}^{*}$ “close” to this ball. Hence, we satisfy this client by using the same fraction $1/k$ of $\boldsymbol{x}^{*}$ (static assignment) while paying a small assignment cost (roughly a constant times the radius of the ball). Since, there are at most $k$ clients in each scenario, and each one is using at most $\boldsymbol{x}^{*}/k$, we need to dedicate only one $\boldsymbol{x}^{*}$ for all clients in $C_{1}$. Formally, we have the following lemma.

Now, consider the set $C_{2}$. By construction, these are clients such that there is not enough supply within a distance $r=5\mathrm{OPT}_{2}/k$ to satisfy half of them. Therefore, intuitively they need to pay “large” distances in the optimal assignment cost if they show up all together in the same scenario. In the following lemma, we show that we can have no more than $k$ of these clients. As we would show later, this would imply that we can dedicate a supply $\boldsymbol{x}^{*}$ to $C_{2}$ and make a static assignment of all the clients $C_{2}$ to this $\boldsymbol{x}^{*}$.

Finally, for clients $C_{3}$, we show that there exists $|C_{3}|/2\alpha$ units of supply “close” to them. In particular, we can multiply these units by $2\alpha$, dedicate them to $C_{3}$ and make a static assignment for $C_{3}$. We have the following lemma.

Similar to the uncapacitated problem, we can solve (Static-SCRFL) efficiently using a compact linear program. In fact, we dualize the inner maximization problem in the objective function of (Static-SCRFL) in the same way as (2). In addition to that, we reformulate the second constraint in (Static-SCRFL) using the same dualization technique as follows: for each $i\in{\cal F}$,

The linear program is given by