A Penalty-Evaporation Heuristic in a Decomposition Method for the Maximum Clique Problem

In this paper, we present a heuristic method to solve the maximum clique problem, based on the concepts of penalty and evaporation. At each iteration, some vertex i is inserted into the current solution (always a clique) and the vertices that are not adjacent to vertex i are removed from the solution. The removed vertices are then penalized in order to reduce their potential of being selected to be inserted in the solution again during the next iterations. This penalty is gradually evaporating to allow vertices to become interesting subsequently. This penalty-evaporation heuristic method is embedded in a decomposition algorithm that restricts the search for a maximum clique to subgraphs, but performs an aggressive exploration of the feasible domain. Numerical results indicate that the penalty-evaporation heuristic method alone is effective and reliable, but the gain in quality obtained when embedding it in the decomposition algorithm is worthy of the additional computing time required.