Lossy Kernelization of Same-Size Clustering

TL;DR

This paper introduces the first lossy polynomial kernel for the equal-size constrained k-median clustering problem, providing a 2-approximate solution and establishing lower bounds that rule out polynomial kernels and PTAS.

Contribution

It presents the first lossy polynomial kernel for the constrained k-median problem and proves lower bounds for exact kernels and PTAS.

Findings

First lossy polynomial kernel for the problem

Achieves a 2-approximate solution

Lower bounds exclude polynomial kernels and PTAS

Abstract

In this work, we study the $k$-median clustering problem with an additional equal-size constraint on the clusters, from the perspective of parameterized preprocessing. Our main result is the first lossy ($2$-approximate) polynomial kernel for this problem, parameterized by the cost of clustering. We complement this result by establishing lower bounds for the problem that eliminate the existences of an (exact) kernel of polynomial size and a PTAS.