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

Abstract

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. 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., minThis 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.

On Generalizations of Network Design Problems with Degree Bounds

“…Another interesting consequence of Theorem 12 is for the robust k-median problem [1]. Here we are given a metric…”

“…Anthony et al [1] gave an O(log m+log k)-approximation algorithm for robust k-median, and showed that it is hard to approximate better than factor two. At first sight this problem may seem unrelated to crossing matroid basis.…”

“…A collection {U 1 , · · · , U t } of vertex-sets is called laminar if for every pair U i , U j in this collection, we have U i ⊆ U j , U j ⊆ U i , or U i ∩ U j = ∅. A (ρ, f (b)) approximation for minimum cost degree bounded problems refers to a solution that (1) has cost at most ρ times the optimum that satisfies the degree bounds, and (2) satisfies the relaxed degree constraints in which a bound b is replaced with a bound f (b).…”

Everywhere-Sparse Spanners via Dense Subgraphs

“…A key property shared by these hierarchies is that they are locally integral; that is, the q-th relaxation in the hierarchy coincides exactly with the convex hull of feasible 0-1 solutions, when both are projected onto any q-dimensional subspace corresponding to q variables in the program. 2 Specifically for Sherali-Adams, the q-th relaxation for a given 0-1 linear program with variables x 1 , . .…”

“…We prove that certain types of "faithful" approximation algorithms for a variant of DkS which we call SMALLEST m-EDGE SUBGRAPH (or SmES) imply approximation algorithms for LD2S, and then show how to construct such an algorithm for SmES; combining these two together yields our improved approximation for LD2S. We seem to be the first to formally define and study SmES, although it has been used in previous work (sometimes implicitly) as the natural minimization version of DkS, see e.g. [35], [25], [2]. In SmES we are given a graph G and a value m, and want to find the subgraph of G with at least m edges that has the fewest vertices.…”