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 $O\left(nm+n{T}_d\right)$ time, where $T_d$ is the time to solve the maximum clique problem in an arbitrary graph on $d$ vertices. The best bound as of now is $T_d=O(2^{d/4})$ by Robson. This shows that the maximum clique problem is solvable in $O(nm)$ time in graphs for which $d\le 4{log}_{2}m+O(1)$ . 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 $2^{O(\sqrt{n})}$ 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.