An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem

TL;DR

This paper presents a new approximation algorithm for the metric uncapacitated facility location problem that improves the approximation ratio to 1.5, approaching the theoretical limit, and extends to multi-level variants.

Contribution

It introduces a (1.6774, 1.3738)-approximation algorithm that improves previous bounds and is the first to reach the approximability limit curve for UFL.

Findings

Achieves a 1.5-approximation ratio for metric UFL.

First algorithm to touch the approximability limit curve.

Improves approximation for multi-level UFL.

Abstract

We obtain a 1.5-approximation algorithm for the metric uncapacitated facility location problem (UFL), which improves on the previously best known 1.52-approximation algorithm by Mahdian, Ye and Zhang. Note, that the approximability lower bound by Guha and Khuller is 1.463. An algorithm is a {\em ($\lambda_f$,$\lambda_c$)-approximation algorithm} if the solution it produces has total cost at most $\lambda_f \cdot F^* + \lambda_c \cdot C^*$, where $F^*$ and $C^*$ are the facility and the connection cost of an optimal solution. Our new algorithm, which is a modification of the $(1+2/e)$-approximation algorithm of Chudak and Shmoys, is a (1.6774,1.3738)-approximation algorithm for the UFL problem and is the first one that touches the approximability limit curve $(\gamma_f, 1+2e^{-\gamma_f})$ established by Jain, Mahdian and Saberi. As a consequence, we obtain the first optimal approximation algorithm for instances dominated by connection costs. When combined with a (1.11,1.7764)-approximation algorithm proposed by Jain et al., and later analyzed by Mahdian et al., we obtain the overall approximation guarantee of 1.5 for the metric UFL problem. We also describe how to use our algorithm to improve the approximation ratio for the 3-level version of UFL.