Cluster Editing: Kernelization based on Edge Cuts

TL;DR

This paper introduces new kernelization techniques for the cluster editing problem based on graph edge cuts, achieving optimal kernel sizes for unweighted cases and improved bounds for weighted cases.

Contribution

It presents novel insights and methods linking cluster editing to edge cuts, leading to a quadratic-time algorithm for constructing optimal kernels.

Findings

Achieves a 2k kernel for weighted cluster editing in O(n^2) time.

Matches the best kernel size for unweighted cluster editing.

Improves the previous quadratic kernel for weighted cluster editing.

Abstract

Kernelization algorithms for the {\sc cluster editing} problem have been a popular topic in the recent research in parameterized computation. Thus far most kernelization algorithms for this problem are based on the concept of {\it critical cliques}. In this paper, we present new observations and new techniques for the study of kernelization algorithms for the {\sc cluster editing} problem. Our techniques are based on the study of the relationship between {\sc cluster editing} and graph edge-cuts. As an application, we present an ${\cal O}(n^2)$-time algorithm that constructs a $2k$ kernel for the {\it weighted} version of the {\sc cluster editing} problem. Our result meets the best kernel size for the unweighted version for the {\sc cluster editing} problem, and significantly improves the previous best kernel of quadratic size for the weighted version of the problem.