Improved Local Search Based Approximation Algorithm for Hard Uniform Capacitated k-Median Problem

TL;DR

This paper presents a local search heuristic that achieves a (3+ε)-approximation for the hard uniform capacitated k-median problem, allowing a controlled violation of the cardinality constraint, and extends to include penalties.

Contribution

It introduces the first local search based approximation algorithm for the problem with cardinality violation, improving upon previous LP-based methods in simplicity and empirical performance.

Findings

Achieves a (3+ε) approximation with 8/3 cardinality violation.

Extends the approach to capacitated k-median with penalties.

Demonstrates practical effectiveness of local search in this context.

Abstract

In this paper, we study the hard uniform capacitated $k$- median problem using local search heuristic. Obtaining a constant factor approximation for the \ckm problem is open. All the existing solutions giving constant-factor approximation, violate at least one of the cardinality and the capacity constraints. All except Koruplou et al are based on LP-relaxation. We give $(3+\epsilon)$ factor approximation algorithm for the problem violating the cardinality by a factor of $8/3 \approx 2.67$. There is a trade-off between the approximation factor and the cardinality violation between our work and the existing work. Koruplou et al gave $(1 + \alpha)$ approximation factor with $(5 + 5/\alpha)$ factor loss in cardinality using local search paradigm. Though the approximation factor can be made arbitrarily small, cardinality loss is at least $5$. On the other hand, we improve upon the results in [capkmGijswijtL2013],[capkmshili2014], [Lisoda2016] in terms of factor-loss though the cardinality loss is more in our case. Also, these results are obtained using LP-rounding, some of them being strengthened, whereas local search techniques are simple to apply and have been shown to perform well in practice via empirical studies. We extend the result to hard uniform capacitated $k$-median with penalties. To the best of our knowledge, ours is the first result for the problem.