The disjoint cliques problem

Abstract

Given a graph G = (V, E), we consider the problem of finding a set of D pairwise disjoint cliques in the graph with maximum overall number of vertices. We determine the computational complexity of this problem restricted to a variety of different graph classes. We give polynomial time algorithms for the problem restricted to interval graphs, cographs, directed path graphs and partial k-trees. In contrast, we show the NP-completeness of this problem for undirected path graphs. Moreover, we investigate a closely related scheduling problem. Given D times units, we look for a sequence of workers w 1 ,...,w K and a partition J 1 ,..., J k of the job set such that J i can be executed by w i within D time units. The goal is to find a sequence with minimum total wage of the workers.