Treewidth: A Useful Marker of Empirical Hardness in Quantified Boolean Logic Encodings

Theoretical studies show that in some combinatorial problems, there is a close relationship between classes of tractable instances and the treewidth (tw) of graphs describing their structure. In the case of satisfiability for quantified Boolean formulas (QBFs), tractable classes can be related to a generalization of treewidth, that we call quantified treewidth (twp). In this paper we investigate the practical relevance of computing twp for problem domains encoded as QBFs. We show that an approximation of twp is a predictor of empirical hardness, and that it is the only parameter among several other candidates which succeeds consistently in being so. We also provide evidence that QBF solvers benefit from a preprocessing phase geared towards reducing twp, and that such phase is a potential enabler for the solution of hard QBF encodings.