LP-based Approximation Algorithms for Capacitated Facility Location

There has been a great deal of recent work on approximation algorithms for facility location problems [9]. We consider the capacitated facility location problem with hard capacities. We are given a set of facilities, F , and a set of clients D in a common metric space. Each facility i has a facility opening cost fi and capacity ui that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set F and assign each client to an open facility so that at most ui clients are assigned to any open facility i. The cost of assigning client j to facility i is given by their distance cij , and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The recent work on facility location problems has come in two varieties: LPbased algorithms, and local search-based algorithms. For the problem described above, no constant approximation algorithm based on LP is known, and in fact, no LP relaxation is known for which the ratio between the optimal integer and fractional values has been bounded by a constant. Surprisingly, constant performance guarantees can still be proved based on local search, but these have the disadvantage that on an instance by instance basis, one never knows anything stronger than the guarantee itself (without resorting to solving an LP relaxation anyway). We present an algorithm that rounds the optimal fractional solution to a natural LP relaxation by using this solution to guide the decomposition of the input into a collection of single-demand-node capacitated facility location problems, which are then solved independently. In the special case that all facility opening costs are equal, we show that our algorithm is a 5-approximation algorithm, thereby also providing the first constant upper bound on the integrality gap of this formulation in this important special case. One salient feature of our algorithm is that it relies on a decomposition of the input into instances of the single-demand capacitated facility location problem; in this way, the algorithm