Instances for the Maximum Clique Problem with Hardness Guarantees
Branch and bound algorithms which use (an estimate on) the chromatic number as the bounding function are reported among the best known exact algorithms for the maximum clique problem. Their average running time is known to be quasi-polynomial and describing families of instances which induce exponential running time for these algorithms is a non-trivial issue. We describe two such families of graphs. We prove that graphs in the first family induce asymptotically higher running time than the average for algorithms whose running time is Ω(c(G)) where c(G) is the number of cliques in G. We also prove that graphs in the second family induce exponential running time for branch and bound algorithms which use the chromatic number as the bounding function.
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