An Efficient Algorithm for Enumerating Pseudo Cliques

The problem of finding dense structures in a given graph is quite basic in informatics including data mining and data engineering. Clique is a popular model to represent dense structures, and widely used because of its simplicity and ease in handling. Pseudo cliques are natural extension of cliques which are subgraphs obtained by removing small number of edges from cliques. We here define a pseudo clique by a subgraph such that the ratio of the number of its edges compared to that of the clique with the same number of vertices is no less than a given threshold value. In this paper, we address the problem of enumerating all pseudo cliques for given a graph and a threshold value. We first show that it seems to be difficult to obtain polynomial time algorithms using straightforward divide and conquer approaches. Then, we propose a polynomial time, polynomial delay in precise, algorithm based on reverse search. We show the efficiency of our algorithm in practice by computational experiments.