A 2-Approximation Algorithm for the Graph 2-Clustering Problem

Abstract

We study a version of the graph 2-clustering problem. In this version, for a given undirected graph, one has to find a nearest 2-cluster graph, i.e., the graph on the same vertex set with exactly 2 nonempty connected components each of which is a complete graph. The distance between two graphs is the number of noncoinciding edges.

The problem under consideration is NP-hard. In 2004, Bansal, Blum, and Chawla presented a simple polynomial time 3-approximation algorithm for the similar correlation clustering problem in which the number of clusters doesn’t exceed 2. In 2008, Coleman, Saunderson, and Wirth presented a 2-approximation algorithm for this problem applying local search to every feasible solution obtained by the 3-approximation algorithm of Bansal, Blum, and Chawla.

Unfortunately, the method of proving the performance guarantee of the Coleman, Saunderson, and Wirth’s algorithm is not suitable for the graph 2-clustering. Coleman, Saunderson, and Wirth used switching technique that allows to reduce clustering any graph to the equivalent problem whose optimal solution is the complete graph, i.e., the cluster graph consisting of the single cluster.

In the graph 2-clustering problem any optimal solution has to consist of exactly 2 clusters, so we need another approximation algorithm and another method of proving a bound on its worst-case behaviour. We present a polynomial time 2-approximation algorithm for the 2-clustering problem on general graphs. In contrast to the proof of Coleman, Saunderson, and Wirth, our proof of the performance guarantee of this algorithm doesn’t use switchings.