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

Abstract

Abstract. This paper proposes new approximate coloring and other related techniques which markedly improve the run time of the branchand-bound algorithm MCR (J. Global Optim., 37, 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. Consequently, it is shown by extensive computational experiments that MCS is remarkably faster than MCR and other existing algorithms. It is faster than the other algorithms by an order 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 graphs. MCS can be faster than MCR by a factor of more than 100,000 for some extremely dense random graphs.

Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover

“…We generally compare the three algorithms for Vertex Cover based on different approaches: B&R, CPLEX and MCS [19]. For instances from Odd Cycle Transversal, we also include the results of the algorithm for directly solving Odd Cycle Transversal by Hüffner [9].…”

“…If there are no such cliques, we add a clique of the single vertex v to C. Since it takes only O(d(v)) time for each vertex v, the algorithm runs in linear time in total. This lower bound is also used in the state-of-the-art branch-and-bound algorithm MCS [19]. MCS computes a clique cover using a more sophisticated strategy to obtain a better lower bound.…”

“…These results are presented in our tables in parentheses. [19] is a state-of-the-art branch-and-bound algorithm for the Maximum Clique problem. We used this algorithm for computing minimum vertex cover by virtually considering complement graphs.…”

An improved bit parallel exact maximum clique algorithm

“…One of the most successful paradigms for fast MCP algorithms is branch-andbound, where a systematic enumeration of maximal diques is pruned by bounding the size of the largest possible clique in the remaining subproblem. A good compromise between computational overhead and tight bounding is obtained through approximate sequential vertex-coloring (usually referred to as SEQ), as in [7][8][9][10][11], and two recent leading algorithms MCS [12] and BB-MaxClique [13].…”

“…Recently (and independently) a very fast new algorithm MCS has been described in [12], which combines the algorithmic improvements proposed in [13] (using conventional data structures) together with a new recoloring strategy (or renumbering as denoted by the authors). Reported tests clearly validate the combined improvements w.r.t.…”