A Systematic Approach to Bound Factor-Revealing LPs and its Application to the Metric and Squared Metric Facility Location Problems

TL;DR

This paper introduces a systematic method for bounding factor-revealing linear programs, applies it to the metric and squared metric facility location problems, and determines tight approximation bounds for algorithms in these settings.

Contribution

It develops a new systematic technique to derive upper bound factor-revealing programs, simplifying the analysis of approximation algorithms for facility location problems.

Findings

Proves no approximation factor better than 2.04 for SMFLP assuming P ≠ NP.

Shows LP-rounding algorithm achieves a 2.04 ratio for SMFLP.

Improves previous analysis of primal-dual algorithms for MFLP and SMFLP.

Abstract

A systematic technique to bound factor-revealing linear programs is presented. We show how to derive a family of upper bound factor-revealing programs (UPFRP), and show that each such program can be solved by a computer to bound the approximation factor of an associated algorithm. Obtaining an UPFRP is straightforward, and can be used as an alternative to analytical proofs, that are usually very long and tedious. We apply this technique to the Metric Facility Location Problem (MFLP) and to a generalization where the distance function is a squared metric. We call this generalization the Squared Metric Facility Location Problem (SMFLP) and prove that there is no approximation factor better than 2.04, assuming P $\neq$ NP. Then, we analyze the best known algorithms for the MFLP based on primal-dual and LP-rounding techniques when they are applied to the SMFLP. We prove very tight bounds for these algorithms, and show that the LP-rounding algorithm achieves a ratio of 2.04, and therefore has the best factor for the SMFLP. We use UPFRPs in the dual-fitting analysis of the primal-dual algorithms for both the SMFLP and the MFLP, improving some of the previous analysis for the MFLP.