A 2 k kernel for the cluster editing problem

Abstract

The cluster editing problem is a decision problem that, for a graph G and a parameter k, determines if one can apply at most k edge insertion/deletion operations on G so that the resulting graph becomes a union of disjoint cliques. The problem has attracted much attention because of its applications in a variety of areas. In this paper, we present a polynomial-time kernelization algorithm for the problem that produces a kernel of size bounded by 2 k. More precisely, we develop an O ( m n ) -time algorithm that, on a graph G of n vertices and m edges and a parameter k, produces a graph G ′ and a parameter k ′ such that k ′ ⩽ k , that G ′ has at most 2 k ′ vertices, and that ( G , k ) is a yes-instance if and only if ( G ′ , k ′ ) is a yes-instance of the cluster editing problem. This improves the previously best kernel of size 4 k for the problem.