A Bit-Parallel Russian Dolls Search for a Maximum Cardinality Clique in a Graph

Finding the clique of maximum cardinality in an arbitrary graph is an NP-Hard problem that has many applications, which has motivated several approaches to solve it exactly despite its difficulty. In this paper, we propose an exact algorithm for the maximum clique problem based on the Russian Dolls Search method. In comparison to a first implementation of this Russian Dolls method from the literature, several improvements are presented. Some of them are adaptations of techniques already employed successfully to this problem, like the use of approximate coloring for pruning and bit-parallel operations. Three different coloring heuristics are tested: the standard greedy and two others from the literature, namely greedy with recoloring and fractional coloring. Other improvements are directly related to the Russian Dolls scheme: the adoption of recursive calls where each subproblem (doll) is solved itself by a Russian Dolls Search and the application of an elimination rule allowing not to generate a significant number of dolls. Results of computational experiments show that the algorithm outperforms the best exact combinatorial algorithms in the literature for the great majority of the instances tested, being more than twice faster in several cases.