A new exact algorithm for the maximum-weight clique problem based on a heuristic vertex-coloring and a backtrack search

In this paper we present an exact algorithm for the maximum-weight clique problem on arbitrary undirected graphs. The algorithm based on a fact that vertices from the same independent set couldn’t be included into the same maximum clique. Those independent sets are obtained from a heuristic vertex coloring where each of them is a color class. Color classes and a backtrack search are used for pruning branches of the maximum-weight clique search tree. Those pruning strategies together result in a very effective algorithm for the maximum-weight clique finding. Computational experiments with random graphs show that the new algorithm works faster than earlier published algorithms; especially on dense graphs.