Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time

TL;DR

This paper presents a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques in sparse graphs, with runtime closely matching the maximum possible number of such cliques based on graph degeneracy.

Contribution

It introduces a modified Bron--Kerbosch algorithm with near-optimal runtime for maximal clique enumeration parameterized by degeneracy, and establishes tight bounds on the number of maximal cliques.

Findings

Algorithm runs in time $O(dn3^{d/3})$

Maximum number of maximal cliques is $(n-d)3^{d/3}$

Algorithm matches worst-case output size bounds

Abstract

The degeneracy of an $n$-vertex graph $G$ is the smallest number $d$ such that every subgraph of $G$ contains a vertex of degree at most $d$. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time $O(dn3^{d/3})$. We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an $n$-vertex graph with degeneracy $d$ (when $d$ is a multiple of 3 and $n\ge d+3$) is $(n-d)3^{d/3}$. Therefore, our algorithm matches the $\Theta(d(n-d)3^{d/3})$ worst-case output size of the problem whenever $n-d=\Omega(n)$.