Certifying Solvers for Clique and Maximum Common (Connected) Subgraph Problems

An algorithm is said to be certifying if it outputs, together with a solution to the problem it solves, a proof that this solution is correct. We explain how state of the art maximum clique, maximum weighted clique, maximal clique enumeration and maximum common (connected) induced subgraph algorithms can be turned into certifying solvers by using pseudo-Boolean models and cutting planes proofs, and demonstrate that this approach can also handle reductions between problems. The generality of our results suggests that this method is ready for widespread adoption in solvers for combinatorial graph problems

Infra-chromatic bound for exact maximum clique search

© 2015 Elsevier Ltd. All rights reserved. Many efficient exact branch and bound maximum clique solvers use approximate coloring to compute an upper bound on the clique number for every subproblem. This technique reasonably promises tight bounds on average, but never tighter than the chromatic number of the graph. Li and Quan, 2010, AAAI Conference, p. 128-133 describe a way to compute even tighter bounds by reducing each colored subproblem to maximum satisfiability problem (MaxSAT). Moreover they show empirically that the new bounds obtained may be lower than the chromatic number. Based on this idea this paper shows an efficient way to compute related >infra-chromatic> upper bounds without an explicit MaxSAT encoding. The reported results show some of the best times for a stand-alone computer over a number of instances from standard benchmarks.Pablo San Segundo is funded by the Spanish Ministry of Economy and Competitiveness (ARABOT: DPI 2010-21247-C02-01) and supervised by CACSA. Alexey Nikolaev and Mikhail Batsyn are supported by Russian Federation Grant 14-41-00039. The authors would like to thank Chu-Min Li and Zhe Quan for the source code of their MaxCLQ algorithm.Peer Reviewe

Constructive Nonsmooth Analysis and Related Topics

XII, 253 p. 35 illus., 11 illus. in color.online