A new upper bound for the maximum weight clique problem

The maximum weight clique problem (MWCP) for a vertex-weighted graph is to find a complete subgraph in which the sum of vertex weights is maximum. The main goal of this paper is to develop an efficient branch-and-bound algorithm to solve the MWCP. As a crucial aspect of branch-and-bound MWCP algorithms is the incorporation of a tight upper bound, we first define a new upper bound for the MWCP, called UBWC, that is based on a novel notion called weight cover. The idea of a weight cover is to compute a set of independent sets of the graph and define a weight function for each independent set so that the weight of each vertex of the graph is covered by such weight functions. We then propose a new branch-and-bound MWCP algorithm called WC-MWC that uses UBWC to reduce the number of branches of the search space that must be traversed by incrementally constructing a weight cover for the graph. Finally, we present experimental results that show that UBWC reduces the search space much more than previous upper bounds, and the new algorithm WC-MWC outperforms some of the best performing exact and heuristic MWCP algorithms on both small/medium graphs and real-world massive graphs.Work supported by the National Natural Science Foundation of China (grants 61272014, 61370183, 61472147 and 61370184), the platforms Matrics of University of Picardie Jules Verne and HPC of Jianghan Univeristy, the Spanish Ministry of Economy and Competitiveness–FEDER (grant TIN2015-71799-C2-1-P).Peer reviewe