On the efficiency of an order-based representation in the clique covering problem

Abstract

Although the (vertex) clique covering problem (CCP) is a classical NP-hard problem, it is still overlooked in the fields of heuristics and evolutionary algorithms. We present two main results concerning this problem. First, we propose a genotype-phenotype mapping algorithm for an order-based representation of the CCP, called greedy clique covering (GCC), and prove that for an arbitrary graph, there is a permutation, for which GCC constructs the optimal solution. Although the greedy graph coloring can also be used as genotype-phenotype mapping, we show that GCC is much more efficient for sparse graphs. Secondly, we adapt a mutation-based metaheuristic algorithm using the order-based representation - iterated greedy (IG), to solve the CCP. On sparse graphs with planted cliques, we provide empirical evidence that IG outperforms an exact algorithm. This result is supported by a runtime analysis of IG on several subclasses of graphs with planted cliques. We include experimental results of IG on random graphs, several DIMACS instances and social graphs. Its comparison to the related existing approaches shows that our IG algorithm outperforms the standard approaches in almost all instances.