A new decomposition technique for maximal clique enumeration for sparse graphs

Given a graph, we are interested in the question of finding all its maximal cliques. This question models the community detection problem and has been extensively studied. Here, we approach it under the light of an important graph parameter. The degeneracy of a graph G is the smallest integer k such that every subgraph of G contains a vertex of degree at most k. Using a new decomposition technique, we present two output sensitive algorithms for the maximal clique enumeration problem. The first one has enumeration time depending only on the degeneracy of the graph. This is the first such result in the literature. This algorithm requires that the cliques are stored in memory. Thus, we propose a second one, which has enumeration time depending on the degeneracy and the maximum degree, and which only requires memory polynomial in the degeneracy of the graph (besides the space to store the graph itself). Then we show that this algorithm can be easily parallelized. As a by-product of our decomposition technique, we propose new algorithms for the maximum clique and p-clique problems as well as an approximation algorithm counting maximal cliques, with approximation ratio the maximum clique size.