A Plant Location Guide for the Unsure: Approximation Algorithms for Min-Max Location Problems

• 成果类型: Article
• 署名作者: Anthony, Barbara; Goyal, Vineet; Gupta, Anupam; Nagarajan, Viswanath
• 署名单位: Massachusetts Institute of Technology (MIT); Carnegie Mellon University; International Business Machines (IBM); IBM USA
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.1090.0428
• 发表日期: 2010
• 页码: 79-101
• 关键词: dense k-subgraph; Facility Location; bounds

摘要

This paper studies an extension of the k-median problem under uncertain demand. We are given an n-vertex metric space (V, d) and m client sets {S-i subset of V}(i=1)(m). The goal is to open a set of k facilities F such that the worst-case connection cost over all the client sets is minimized, i.e., min(F subset of V,vertical bar F vertical bar=k) max(i is an element of[m]){Sigma(d(j,f))(j is an element of si)}, where for any F subset of V, d(j, F) = min(f is an element of F) d(j, f). This is a min-max or robust version of the k-median problem. Note that in contrast to the recent papers on robust and stochastic problems, we have only one stage of decision-making where we select a set of k facilities to open. Once a set of open facilities is fixed, each client in the uncertain client-set connects to the closest open facility. We present a simple, combinatorial O (log n+log m)-approximation algorithm for the robust k-median problem that is based on reweighting/Lagrangean-relaxation ideas. In fact, we give a general framework for (minimization) k-facility location problems where there is a bound on the number of open facilities. We show that if the location problem satisfies a certain projection property, then both the robust and stochastic versions of the location problem admit approximation algorithms with logarithmic ratios. We use our framework to give the first approximation algorithms for robust and stochastic versions of several location problems such as k-tree, capacitated k-median, and fault-tolerant k-median.