LP-Based Algorithms for Capacitated Facility Location

Abstract

Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no linear programming relaxation was known to efficiently approximate the problem.In this paper, we present a linear programming relaxation with constant integrality gap for capacitated facility location. We demonstrate that the fundamental theories of multi-commodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding.

Improved approximation for tree augmentation: saving by rewiring

“…Ellipsoid-based techniques with such partial separation oracles have been employed successfully in prior work, including in the context of facility location [3,18,2], and very recently, Nutov [20] presented an elegant application of it by showing that a certain LP has integrality gap below 2 for weighted TAP with bounded ratio between highest and lowest cost.…”

“…These constraints imply the cut-LP constraints. 2 Moreover, one can efficiently separate over Π CG T (see [9]). A key property of Π CG T shown by Fiorini et al [9], is that points in Π CG T can be rounded losslessly on a particular type of instance, as summarized below.…”

“…Hence, for large enough k, any integrality gap instance is a k-wide TAP. 2 The fact that the cut-LP constraint for any edge e ∈ T is implied by Π CG T , and hence Π CG T ⊆ ΠT , follows by considering the set S corresponding to the vertex set of one of the two connected components of T when removing the edge e. Corollary 6 ([9]). Given a point x ∈ Π CG T , we can compute in polynomial time a solution A such that…”

“…In this section, we show a technical statement, stated as Lemma 16 below, about the maximizer of a function used in Lemma 15, which completes the proof of Lemma 15. Let f 1 (x, y) = 1 − x − y, f 2 (x, y) = x − (x−y) 2 4(1−y) and f (x, y) = min{f 1 (x, y), f 2 (x, y)}. Using this notation, the maximization problem we consider is…”

Easy capacitated facility location problems, with connections to lot-sizing

“…In this case, a simple local search algorithm initially given by Korupolu, Plaxton, and Rajaraman [19] has been shown by Aggarwal, Louis, Bansal, Garg, Gupta, Gupta, and Jain [1] to be a (3 + ǫ)-approximation algorithm in the uniform case; a less simple local search algorithm is a (5 + ǫ)-approximation algorithm for the non-uniform case [5]. An, Singh, and Svensson [2] have given a linear programming relaxation of the problem such that an LP rounding algorithm is a 316-approximation algorithm for the problem. A long line of research has given approximation algorithms with small performance guarantees in the case of the metric uncapacitated facility location problem, in which U i = ∞ for all facilities i; the best currently known algorithm is a 1.488-approximation algorithm due to Li [21].…”