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A clique in an undirected graph G= (V, E) is a subset V' V of vertices, eachpair of which is connected by an edge in E. The clique problem is anoptimization problem of finding a clique of maximum size in graph. The cliqueproblem is NP-Complete. We have succeeded in developing a fast algorithm formaximum clique problem by employing the method of verification and elimination.For a graph of size N there are 2N sub graphs, which may be cliques and henceverifying all of them, will take a long time. Idea is to eliminate a majornumber of sub graphs, which cannot be cliques and verifying only the remainingsub graphs. This heuristic algorithm runs in polynomial time and executessuccessfully for several examples when applied to random graphs and DIMACSbenchmark graphs.

A Fast Heuristic Algorithm Based on Verification and Elimination Methods for Maximum Clique Problem