Approximation Algorithms for Graph Clustering Problems with Clusters of Bounded Size

Abstract

In the cluster editing problem, one has to partition the set of vertices of a graph into pairwise disjoint subsets (called clusters) minimizing the number of edges between clusters and the number of missing edges within clusters. We consider a version of the problem in which cluster sizes are bounded from above by a positive integer \( s \). This problem is NP-hard for any fixed \( s \geqslant 3 \). We propose polynomial-time approximation algorithms for this version of the problem. Their performance guarantees equal \( 5/3 \) for the case \( s = 3 \) and \( 2 \) for \( s = 4 \). We also show that the cluster editing problem is APX-hard for the case of \( s = 3 \).