Centrality of Trees for Capacitated k-Center

TL;DR

This paper presents a simple, LP-based 9-approximation algorithm for the capacitated k-center problem, bridging the gap with uncapacitated cases and improving understanding of capacity constraints in network location problems.

Contribution

It introduces a versatile tree-based approach and provides the first near-tight approximation guarantee for the capacitated k-center problem using standard LP relaxation.

Findings

Achieves a 9-approximation guarantee for the capacitated k-center problem.

Nears the integrality gap of 7, 8, or 9 after preprocessing.

Provides improved algorithms for variants with uniform capacities.

Abstract

There is a large discrepancy in our understanding of uncapacitated and capacitated versions of network location problems. This is perhaps best illustrated by the classical k-center problem: there is a simple tight 2-approximation algorithm for the uncapacitated version whereas the first constant factor approximation algorithm for the general version with capacities was only recently obtained by using an intricate rounding algorithm that achieves an approximation guarantee in the hundreds. Our paper aims to bridge this discrepancy. For the capacitated k-center problem, we give a simple algorithm with a clean analysis that allows us to prove an approximation guarantee of 9. It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either 7, 8 or 9. The algorithm proceeds by first reducing to special tree instances, and then solves such instances optimally. Our concept of tree instances is quite versatile, and applies to natural variants of the capacitated k-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are uniform.