A Simple and Faster Branch-and-Bound Algorithm for Finding a Maximum Clique with Computational Experiments

Abstract

SUMMARYMany problems can be formulated as maximum clique problems. Hence, it is highly important to develop algorithms that can find a maximum clique very fast in practice. We propose new approximate coloring and other related techniques which markedly improve the run time of the branch-and-bound algorithm MCR (J. Global Optim., 37, pp.95-111, 2007), previously shown to be the fastest maximum-clique-finding algorithm for a large number of graphs. The algorithm obtained by introducing these new techniques in MCR is named MCS. It is shown that MCS is successful in reducing the search space quite efficiently with low overhead. Extensive computational experiments confirm the superiority of MCS over MCR and other existing algorithms. It is faster than the other algorithms by orders of magnitude for several graphs. In particular, it is faster than MCR for difficult graphs of very high density and for very large and sparse graphs, even though MCS is not designed for any particular type of graph. MCS can be faster than MCR by a factor of more than 100,000 for some extremely dense random graphs. This paper demonstrates in detail the effectiveness of each new techniques in MCS, as well as the overall contribution. key words : maximum clique, branch-and-bound, approximate coloring, computational experiments

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

“…2 for an illustration. Procedure Re-NUMBER is a central part of MCS and is shown to be quite effective [25], [32], [34]. Some of its variations are also employed in this paper.…”

“…5 in Ref. [34] for an illustration.) The above-mentioned reconstruction of the adjacency matrix is to localize the memory usage, and it results in a more effective use of the cache memory.…”

“…The algorithm obtained by combining all the techniques described in this section is named MCS [25], [32], [34].…”

“…Furthermore, much faster algorithms are in great demand so that much more practical problems can be solved. Along this line, Tomita et al developed a series of branch-and-bound algorithms MCQ [29], MCR [31] and MCS [32], [34] among others that run fast in practice. It was experimentally shown that MCS is relatively fast for many instances tested.…”

“…First, we turn back to our original MCS [25] that employs an approximation algorithm for the maximum clique problem at the beginning in order to obtain an initial lower bound on the size of a maximum clique, as noted at Concluding Remarks in Ref. [34]. We choose here another approximation algorithm called k-opt local search (KLS for short) by Katayama et al [11] that runs in quite a short time.…”