A local core number based algorithm for the maximum clique problem

The maximum clique problem (MCP) is to determine a complete subgraph of maximum cardinality in a graph. MCP is a fundamental problem in combinatorial optimization and is noticeable for its wide range of applications. In this paper, we present two branch-and-bound exact algorithms for finding a maximum clique in an undirected graph. Many efficient exact branch and bound maximum clique algorithms use approximate coloring to compute an upper bound on the clique number but, as a new pruning strategy, we show that local core numbers are more efficient. Moreover, instead of neighbors set of a vertex, our search area is restricted to a subset of the set in each subproblem which speeds up clique finding process. This subset is based on the core of the vertices of a given graph. We improved the MCQ and MaxCliqueDyn algorithms with respect to the new pruning strategy and search area restriction. Experimental results demonstrate that the improved algorithms outperform the previous well-known algorithms for many instances when applied to DIMACS benchmark and random graphs.