Reporting maximal cliques: new insights into an old problem

Abstract

Reporting maximal cliques of a graph is a fundamental problem arising many areas. Surprisingly, this problem is often tackled resorting to the old Bron-Kerbosch algorithm (1973), or its recent variant by I. Koch (2001). Both algorithms suffer from a poor output sensitivity and worst-cases. In this context, this paper makes three contributions. First, we show that a slight modification of the greedy pivoting strategy used by I. Koch allows one to get rid of worst-cases, also improving overall performances. Second, exploiting the recursive structure of non maximal cliques, we show the pivoting strategy developed by I. Koch is a particular case of a more general optimization strategy based on the concept of {\em dominated} nodes. Using different instantiations of this concept, we design four modified Bron-Kerbosch algorithms, with better output-sensitivity. Third, we discuss implementation issues and provide a detailed experimental study on random graphs. The bottom-line of this study is the investigation of the trade-off between output sensitivity and the overhead associated to the identification of dominated nodes.