A heuristic for listing almost-clique minimal separators of a graph

Abstract

Bodlaender and Koster (Discrete Mathematics 2006) introduced the notion of almost-clique separators in the context of computing the treewidth $\tw(G)$ of a given graph $G$. A separator $S \subseteq V(G)$ of $G$ is an \emph{almost-clique separator} if $S \setminus \{v\}$ is a clique of $G$ for some $v \in S$. $S$ is a \emph{minimal separator} if $S$ has at least two full components, where a full component of $S$ is a connected component $C$ of $G \setminus S$ such that $N_G(C) = S$. They observed that if $S$ is an almost-clique minimal separator of $G$ then $\tw(G \cup K(S)) = \tw(G)$, where $K(S)$ is the complete graph on vertex set $S$: in words, filling an almost-clique minimal separator into a clique does not increase the treewidth. Based on this observation, they proposed a preprocessing method for treewidth computation, a fundamental step of which is to find a preferably maximal set of pairwise non-crossing almost-clique minimal separators of a graph. In this paper, we present a heuristic for this step, which is based on the following empirical observation. For graph $G$ and a minimal triangulation $H$ of $G$, let $\QQ(H, G)$ denote the set of all almost-clique minimal separators of $G$ that are minimal separators of $H$. Note that since the minimal separators of $H$ are pairwise non-crossing, so are those in $\QQ(H, G)$. We observe from experiments that $\QQ(H, G)$ is remarkably close to maximal, especially when the minimal triangulation $H$ is computed by an algorithm aiming for small treewidth. This observation leads to an efficient implementation of the preprocessing method proposed by Bodlaender and Koster. Experiments on instances from PACE 2017 and other sources show that this implementation is extremely fast and effective for graphs of practical interest.