Fabián A. Chudak

Improved approximation algorithms for capacitated facility location problems

Abstract.In a surprising result, Korupolu, Plaxton, and Rajaraman [13] showed that a simple local search heuristic for the capacitated facility location problem (CFLP) in which the service costs obey the triangle inequality produces a solution in polynomial time which is within a factor of 8+ε of the value of an optimal solution. By simplifying their analysis, we are able to show that the same heuristic produces a solution which is within a factor of 6(1+ε) of the value of an optimal solution. Our simplified analysis uses the supermodularity of the cost function of the problem and the integrality of the transshipment polyhedron.Additionally, we consider the variant of the CFLP in which one may open multiple copies of any facility. Using ideas from the analysis of the local search heuristic, we show how to turn any α-approximation algorithm for this variant into a polynomial-time algorithm which, at an additional cost of twice the optimum of the standard CFLP, opens at most one additional copy of any facility. This allows us to transform a recent 2-approximation algorithm of Mahdian, Ye, and Zhang [17] that opens many additional copies of facilities into a polynomial-time algorithm which only opens one additional copy and has cost no more than four times the value of the standard CFLP.

Near-optimal solutions to large-scale facility location problems

We investigate the solution of large-scale instances of the capacitated and uncapacitated facility location problems. Let n be the number of customers and m the number of potential facility sites. For the uncapacitated case we solved instances of size mxn=3000x3000; for the capacitated case the largest instances were 1000x1000. We use heuristics that produce a feasible integer solution and use a Lagrangian relaxation to obtain a lower bound on the optimal value. In particular, we present new heuristics whose gap from optimality was generally below 1%. The heuristics combine the volume algorithm and randomized rounding. For the uncapacitated facility location problem, our computational experiments show that our heuristic compares favorably against DUALOC.

Approximate k-MSTs and k-Steiner Trees via the Primal-Dual Method and Lagrangean Relaxation

We consider the problem of computing the minimum-cost tree spanning at least k vertices in an undirected graph. Garg [10] gave two approximation algorithms for this problem. We show that Gargs algorithms can be explained simply with ideas introduced by Jain and Vazirani for the metric uncapacitated facility location and k-median problems [15], in particular via a Lagrangean relaxation technique together with the primal-dual method for approximation algorithms. We also derive a constant-factor approximation algorithm for the k-Steiner tree problem using these ideas, and point out the common features of these problems that allow them to be solved with similar techniques.

Improved Approximation Algorithms for Capacitated Facility Location Problems

In a recent surprising result, Korupolu, Plaxton, and Rajaraman [10,11] showed that a simple local search heuristic for the capacitated facility location problem (CFLP) in which the service costs obey the triangle inequality produces a solution in polynomial time which is within a factor of 8 + Ɛ of the value of an optimal solution. By simplifying their analysis, we are able to show that the same heuristic produces a solution which is within a factor of 6(1 + Ɛ) of the value of an optimal solution. Our simplified analysis uses the supermodularity of the cost function of the problem and the integrality of the transshipment polyhedron. Additionally, we consider the variant of the CFLP in which one may open multiple copies of any facility. Using ideas from the analysis of the local search heuristic, we show how to turn any α-approximation algorithm for this variant into one which, at an additional cost of twice the optimum of the standard CFLP, opens at most one additional copy of any facility. This allows us to transform a recent 3-approximation algorithm of Chudak and Shmoys [5] that opens many additional copies of facilities into a polynomial-time algorithm which only opens one additional copy and has cost no more than five times the value of the standard CFLP.

Approximate k -MSTs and k -Steiner trees via the primal-dual method and Lagrangean relaxation

Abstract.Garg [10] gives two approximation algorithms for the minimum-cost tree spanning k vertices in an undirected graph. Recently Jain and Vazirani [15] discovered primal-dual approximation algorithms for the metric uncapacitated facility location and k-median problems. In this paper we show how Garg’s algorithms can be explained simply with ideas introduced by Jain and Vazirani, in particular via a Lagrangean relaxation technique together with the primal-dual method for approximation algorithms. We also derive a constant factor approximation algorithm for the k-Steiner tree problem using these ideas, and point out the common features of these problems that allow them to be solved with similar techniques.

Solving Large Scale Uncapacitated Facility Location Problems

We investigate the solution of instances of the uncapacitated facility location problem with at most 3000 potential facility locations and similar number of customers. We use heuristics that produce a feasible integer solution and a lower bound on the optimum. In particular, we present a new heuristic whose gap from optimality was generally below 1%. The heuristic combines the volume algorithm and a recent approximation algorithm based on randomized rounding. Our computational experiments show that our heuristic compares favorably against DUALOC.