Comparison of Clique-Listing Algorithms

This paper compares several published algorithms which list all of the maximal cliques of a graph. A clique is a complete subgraph, i.e., a set of vertices and edges such that every pair of vertices is joined by an edge. A maximal clique is a clique which is not a proper subgraph of a larger clique. A closely related concept is that of a maximal independent set (MIS), where every pair of vertices is lacking an edge. An algorithm which lists maximal cliques for a graph G can also list MISs of graph G, if the input is changed from G to the complement of G, i.e., each edge is converted to a nonedge and vice versa. Many algorithms have been developed for finding all of the maximal cliques or MISs of a graph, but few papers compare more than a couple of these algorithms experimentally. Some papers give theoretical bounds on the complexity of algorithms, but improved theoretical bounds do not always translate into improved practical performance. This state of affairs makes it difficult for a researcher looking for the fastest algorithm to list all the maximal cliques of a graph. In this paper we compare five cliqueor MIS-listing algorithms, and show that variations of the Bron and Kerbosch algorithm appear to be the fastest for random graphs and a type of graph which arises in physical mapping of genomes.