An approximation algorithm for Uniform Capacitated k-Median problem with 1 + {\epsilon} capacity violation

TL;DR

This paper presents a constant factor approximation algorithm for the Uniform Capacitated k-Median problem that only violates capacity constraints by a small factor, using a configuration LP and dependent rounding.

Contribution

It introduces a novel approximation algorithm that achieves capacity violation only by a factor of 1+epsilon, improving upon previous methods that also violated the number of facilities.

Findings

Achieves constant factor approximation with 1+epsilon capacity violation.

Uses configuration LP and dependent rounding techniques.

Improves upon prior algorithms violating multiple constraints.

Abstract

We study the Capacitated k-Median problem, for which all the known constant factor approximation algorithms violate either the number of facilities or the capacities. While the standard LP-relaxation can only be used for algorithms violating one of the two by a factor of at least two, Shi Li [SODA'15, SODA'16] gave algorithms violating the number of facilities by a factor of 1+{\epsilon} exploring properties of extended relaxations. In this paper we develop a constant factor approximation algorithm for Uniform Capacitated k-Median violating only the capacities by a factor of 1+{\epsilon}. The algorithm is based on a configuration LP. Unlike in the algorithms violating the number of facilities, we cannot simply open extra few facilities at selected locations. Instead, our algorithm decides about the facility openings in a carefully designed dependent rounding process.