On minimization of the number of branches in branch-and-bound algorithms for the maximum clique problem

• Published in: Computers & operations research Vol. 84; pp. 1 - 15
• Main Authors: Li, Chu-Min, Jiang, Hua, Manyà, Felip
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01-08-2017; Pergamon Press Inc; Elsevier
• Subjects: Algorithms; Branch & bound algorithms; Branch-and-bound; Branching ordering; Computer Science; Incremental MaxSAT Reasoning; Maximum clique problem; Operations research; Optimization; Searching; Strategy; Branch-and-bound; Maximum clique problem; Incremental MaxSAT Reasoning; Branching ordering
• ISSN: 0305-0548; 1873-765X; 0305-0548
• DOI: 10.1016/j.cor.2017.02.017

•Incremental MaxSAT reasoning is effective to reduce branches in MaxClique algorithms.•An efficient static strategy for reducing branches in MaxClique algorithms is proposed.•Combining a static and a dynamic strategy highly improves BnB MaxClique algorithms.•The new highly efficient MaxClique algorithms DoMC2, SoMC2 and MoMC are described.
When searching for a maximum clique in a graph G, branch-and-bound algorithms in the literature usually focus on the minimization of the number of branches generated at each search tree node. We call dynamic strategy this minimization without any constraint, because it induces a dynamic vertex ordering in G during the search. In this paper, we introduce a static strategy that minimizes the number of branches subject to the constraint that a static vertex ordering in G must be kept during the search. We analyze the two strategies and show that they are complementary. From this complementarity, we propose a new algorithm, called MoMC, that combines the strengths of the two strategies into a single algorithm. The reported experimental results show that MoMC is generally better than the algorithms implementing a single strategy.