Solving maximum clique in sparse graphs: an $O(nm+n2^{d/4})$ algorithm for $d$-degenerate graphs

Abstract

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.