Efficiently Enumerating Minimal Triangulations

TL;DR

This paper introduces an efficient algorithm for enumerating all minimal graph triangulations and proper tree decompositions in incremental polynomial time, enabling high-quality decompositions for various applications.

Contribution

The authors develop a novel incremental polynomial-time enumeration algorithm for minimal triangulations and proper tree decompositions, adaptable with existing triangulation methods.

Findings

Algorithm outperforms existing methods in quality measures

Able to generate many high-quality decompositions

Effective on real-world data from multiple fields

Abstract

We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions, in incremental polynomial time, where "proper" means that the tree decomposition cannot be improved by removing or splitting a bag. The algorithm can incorporate any method for (ordinary, single result) triangulation or tree decomposition, and can serve as an anytime algorithm to improve such a method. We describe an extensive experimental study of an implementation on real data from different fields. Our experiments show that the algorithm improves upon central quality measures over the underlying tree decompositions, and is able to produce a large number of high-quality decompositions.