Output-Sensitive Enumeration of Potential Maximal Cliques in Polynomial Space

TL;DR

The paper tackles the problem of efficiently enumerating all potential maximal cliques (PMCs) in a graph, a task central to computing treewidth and related graph parameters. Building on the Bouchitté–Todinca framework, it addresses the exponential-space drawback by (i) introducing a duplication-avoidance modification to prevent repeated PMCs, and (ii) presenting a depth-first, polynomial-space enumeration that outputs PMCs on the fly, preserving output-polynomial time. The authors prove correctness of the duplicate-free variant and establish a polynomial-space, on-the-fly enumeration algorithm with a final bound of $O(n^9 m^2 | abla_G|^4)$ time and $O(n^3)$ space (and $O(n^9 m^2 | abla_G|^4)$ in the PMC-counted expression), enabling practical PMC enumeration. This advance has practical implications for treewidth computations and related problems such as maximum-weight independent sets in specific graph classes, offering a pathway to more scalable applications.

Abstract

A set of vertices in a graph forms a potential maximal clique if there exists a minimal chordal completion in which it is a maximal clique. Potential maximal cliques were first introduced as a key tool to obtain an efficient, though exponential-time algorithm to compute the treewidth of a graph. As a byproduct, this allowed to compute the treewidth of various graph classes in polynomial time. In recent years, the concept of potential maximal cliques regained interest as it proved to be useful for a handful of graph algorithmic problems. In particular, it turned out to be a key tool to obtain a polynomial time algorithm for computing maximum weight independent sets in $P_5$-free and $P_6$-free graphs (Lokshtanov et al., SODA `14 and Grzeskik et al., SODA `19. In most of their applications, obtaining all the potential maximal cliques constitutes an algorithmic bottleneck, thus motivating the question of how to efficiently enumerate all the potential maximal cliques in a graph $G$. The state-of-the-art algorithm by Bouchitté \& Todinca can enumerate potential maximal cliques in output-polynomial time by using exponential space, a significant limitation for the size of feasible instances. In this paper, we revisit this algorithm and design an enumeration algorithm that preserves an output-polynomial time complexity while only requiring polynomial space.

Figures (1)

• Figure 1: (a) a simple graph on 4 vertices. (b) Two possible minimal triangulations of the graph (dashed line or dotted line). (c) Two minimal separators of the graph: $\{1,3\}$ and $\{2,4\}$. The potential maximal cliques of (a) include all the maximal cliques of all the different minimal triangulations, that is to say, $\{1,2,3\}$ and $\{1,3,4\}$ from the dotted triangulation and $\{1,2,4\}$, $\{2,3,4\}$ from the dashed triangulation.

Theorems & Definitions (16)

• definition 1: Full Component
• theorem 1: Characterization of PMCs bouchitte2001treewidth
• proposition 1
• proof
• lemma 1
• proof
• lemma 2
• proof
• proposition 2
• proof