Solving maximum clique in sparse graphs: an O(nm+n2d/4) algorithm for d-degenerate graphs
We describe an algorithm for the maximum clique problem that is parameterized by the graph’s degeneracy d. The algorithm runs in O(nm+nTd) time, where Td is the time to solve the maximum clique problem in an arbitrary graph on d vertices. The best bound as of now is Td=O(2d/4)by Robson. This shows that the maximum clique problem is solvable in O(nm) time in graphs for which d≤4log2m+O(1). The analysis of the algorithm’s runtime is simple; the algorithm is easy to implement when given a subroutine for solving maximum clique in small graphs; it is easy to parallelize. In the case of Bianconi-Marsili power-law random graphs, it runs in 2O(√n)time with high probability. We extend the approach for a graph invariant based on common neighbors, generating a second algorithm that has a smaller exponent at the cost of a larger polynomial factor.