The Shape of the Search Tree for the Maximum Clique Problem, and the Implications for Parallel Branch and Bound

Finding a maximum clique in a given graph is one of the fundamental NP-hard problems. We compare two multi-core threaded parallel adaptations of a state-of-the-art branch and bound algorithm for the maximum clique problem, and provide a novel explanation as to why they are successful. We show that load balance is sometimes a problem, but that the interaction of parallel search order and the most likely location of solutions within the search space is often the dominating consideration. We use this explanation to propose a new low-overhead, scalable work splitting mechanism. Our approach uses explicit early diversity to avoid strong commitment to the weakest heuristic advice, and late resplitting for balance.