A Much Faster Algorithm for Finding a Maximum Clique with Computational Experiments

We present further improvements to a branch-and-bound maximum-clique-finding algorithm MCS (WALCOM 2010, LNCS 5942, pp.191–203) that was shown to be fast. First, we employ a variant of an efficient approximation algorithm KLS for finding a maximum clique. Second, we make use of appropriate sorting of vertices only near the root of the search tree. Third, we employ a lightened approximate coloring mainly near the leaves of the search tree. A new algorithm obtained from MCS with the above improvements is named k5 MCT. It is shown that k5 MCT is much faster than MCS by extensive computational experiments. In particular, k5 MCT is shown to be faster than MCS for gen400 p0.9 75, gen400 p0.9 65 and gen400 p0.9 55 by over 81,000, 39,000 and 19,000 times, respectively.