LP-based approximation algorithms for capacitated facility location

• Published in: Mathematical programming Vol. 131; no. 1-2; pp. 365 - 379
• Main Authors: Levi, Retsef, Shmoys, David B., Swamy, Chaitanya
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.02.2012; Springer; Springer Nature B.V
• Subjects: Algorithms; Applied sciences; Approximation; Calculus of Variations and Optimal Control; Optimization; Clients; Combinatorics; Cost engineering; Costs; Exact sciences and technology; Flows in networks. Combinatorial problems; Full Length Paper; Guarantees; Integer programming; Linear programming; Location analysis; Logistics; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Operations research; Optimization; Position (location); Scripts; Studies; Theoretical; logistics; Mathematical Programming; 90B06 Operations Research; transportation; Costs; Operations research; Metric space; Rounding error; Approximation algorithm; Location problem; Facility location; Partitioning; Capacity; Opening; Relaxation method; Mathematical programming; Lp approximation; Lower bound; Local search; Linear programming; Standards; Optimal solution; NP hard problem; Open systems; Problem solving; Capacity constraint; Logistics
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-010-0380-8

In the capacitated facility location problem with hard capacities, we are given a set of facilities, , and a set of clients in a common metric space. Each facility i has a facility opening cost f i and capacity u i that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set and assign each client to an open facility so that at most u i clients are assigned to any open facility i . The cost of assigning client j to facility i is given by the distance c ij , and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The only known approximation algorithms that deliver solutions within a constant factor of optimal for this NP-hard problem are based on local search techniques. It is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound (or indeed, to prove a constant integrality gap for any LP relaxation). We make progress on this question by giving a 5-approximation algorithm for the special case in which all of the facility costs are equal, by rounding the optimal solution to the standard LP relaxation. One notable aspect of our algorithm is that it relies on partitioning the input into a collection of single-demand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way.