Fault-Tolerant Facility Location: a randomized dependent LP-rounding algorithm

TL;DR

This paper introduces a new randomized LP-rounding algorithm with a 1.725-approximation ratio for the metric Fault-Tolerant Uncapacitated Facility Location problem, improving previous bounds and pioneering the use of dependent rounding in this domain.

Contribution

It presents the first application of dependent rounding in facility location, with a novel hierarchical clustering scheme that enhances approximation quality.

Findings

Achieved a 1.725-approximation ratio, better than previous 2.076-approximation.

Developed a hierarchical clustering scheme for facility location.

Extended analysis techniques for dependent rounding in this context.

Abstract

We give a new randomized LP-rounding 1.725-approximation algorithm for the metric Fault-Tolerant Uncapacitated Facility Location problem. This improves on the previously best known 2.076-approximation algorithm of Swamy & Shmoys. To the best of our knowledge, our work provides the first application of a dependent-rounding technique in the domain of facility location. The analysis of our algorithm benefits from, and extends, methods developed for Uncapacitated Facility Location; it also helps uncover new properties of the dependent-rounding approach. An important concept that we develop is a novel, hierarchical clustering scheme. Typically, LP-rounding approximation algorithms for facility location problems are based on partitioning facilities into disjoint clusters and opening at least one facility in each cluster. We extend this approach and construct a laminar family of clusters, which then guides the rounding procedure. It allows to exploit properties of dependent rounding, and provides a quite tight analysis resulting in the improved approximation ratio.