Graph Width Measures for CNF-Encodings with Auxiliary Variables

TL;DR

This paper investigates the limitations and relationships of graph width measures in CNF-formulas with auxiliary variables, revealing significant restrictions on expressivity and strong links between different width measures.

Contribution

It demonstrates that bounding width measures reduces expressivity, and establishes tight relationships between optimal widths across different measures in auxiliary-variable encodings.

Findings

Bounding width measures restrict expressivity to low communication complexity.

Optimal encoding widths are strongly linked within two main classes of measures.

Width differences between classes are at most logarithmic in the number of variables.

Abstract

We consider bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNF-formulas. Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNF-formulas. Here we consider the expressivity of these formulas in the model of clausal encodings with auxiliary variables. We first show that bounding the width for many of the measures from the literature leads to a dramatic loss of expressivity, restricting the formulas to such of low communication complexity. We then show that the width of optimal encodings with respect to different measures is strongly linked: there are two classes of width measures, one containing primal treewidth and the other incidence cliquewidth, such that in each class the width of optimal encodings only differs by constant factors. Moreover, between the two classes the width differs at most by a factor logarithmic in the number of variables. Both these results are in stark contrast to the setting without auxiliary variables where all width measures we consider here differ by more than constant factors and in many cases even by linear factors.