Reducing the Branching in a Branch and Bound Algorithm for the Maximum Clique Problem

Finding the largest clique in a given graph is one of the fundamental NP-hard problems. We take a widely used branch and bound algorithm for the maximum clique problem, and discuss an alternative way of understanding the algorithm which closely resembles a constraint model. By using this view, and by taking measurements inside search, we provide a new explanation for the success of the algorithm: one of the intermediate steps, by coincidence, often approximates a “smallest domain first” heuristic. We show that replacing this step with a genuine “smallest domain first” heuristic leads to a reduced branching factor and a smaller search space, but longer runtimes. We then introduce a “domains of size two first” heuristic, which integrates cleanly into the algorithm, and which both reduces the size of the search space and gives a reduction in runtimes.