Ranked Enumeration of Minimal Triangulations

TL;DR

This paper introduces the first ranked enumeration algorithm for minimal triangulations, enabling the generation of high-quality tree decompositions for various applications under broad cost functions.

Contribution

It presents a novel algorithm for enumerating proper tree decompositions with polynomial delay, generalizing previous methods and applicable to a wide class of cost functions.

Findings

Algorithm achieves polynomial delay under poly-MS assumption.

Experimental results show effectiveness on diverse graph types.

Method outperforms existing approaches in quality and efficiency.

Abstract

A tree decomposition of a graph facilitates computations by grouping vertices into bags that are interconnected in an acyclic structure, hence their importance in a plethora of problems such as query evaluation over databases and inference over probabilistic graphical models. The relative benefit from different tree decompositions is measured by diverse (sometime complex) cost functions that vary from one application to another. For generic cost functions like width and fill-in, an optimal tree decomposition can be efficiently computed in some cases, notably when the number of minimal separators is bounded by a polynomial (due to Bouchitte and Todinca), we refer to this assumption as "poly-MS." To cover the variety of cost functions in need, it has recently been proposed to devise algorithms for enumerating many decomposition candidates for applications to choose from using specialized, or even machine-learned, cost functions. We explore the ability to produce a large collection of "high quality" tree decompositions. We present the first algorithm for ranked enumeration of the proper (non-redundant) tree decompositions, or equivalently minimal triangulations, under a wide class of cost functions that substantially generalizes the above generic ones. On the theoretical side, we establish the guarantee of polynomial delay if poly-MS is assumed, or if we are interested in tree decompositions of a width bounded by a constant. We describe an experimental evaluation on graphs of various domains (including join queries, Bayesian networks, treewidth benchmarks and random), and explore both the applicability of the poly-MS assumption and the performance of our algorithm relative to the state of the art.