Relating Proof Complexity Measures and Practical Hardness of SAT

Abstract

Abstract. Boolean satisfiability (SAT) solvers have improved enormously in performance over the last 10-15 years and are today an indispensable tool for solving a wide range of computational problems. However, our understanding of what makes SAT instances hard or easy in practice is still quite limited. A recent line of research in proof complexity has studied theoretical complexity measures such as length, width, and space in resolution, which is a proof system closely related to state-of-the-art conflict-driven clause learning (CDCL) SAT solvers. Although it seems like a natural question whether these complexity measures could be relevant for understanding the practical hardness of SAT instances, to date there has been very limited research on such possible connections. This paper sets out on a systematic study of the interconnections between theoretical complexity and practical SAT solver performance. Our main focus is on space complexity in resolution, and we report results from extensive experiments aimed at understanding to what extent this measure is correlated with hardness in practice. Our conclusion from the empirical data is that the resolution space complexity of a formula would seem to be a more fine-grained indicator of whether the formula is hard or easy than the length or width needed in a resolution proof. On the theory side, we prove a separation of general and tree-like resolution space, where the latter has been proposed before as a measure of practical hardness, and also show connections between resolution space and backdoor sets.

A (Biased) Proof Complexity Survey for SAT Practitioners

“…The most comprehensive study to date of the question if and how hardness with respect to these complexity measures for resolution is correlated with actual hardness as measured by CDCL running time would seem to be [27], but it seems fair to say that the results so far are somewhat inconclusive.…”

Pebble Games, Proof Complexity, and Time-Space Trade-offs

“…Expressed in terms of formula size the separation one obtains is in the constant multiplicative factor in front of the logarithmic space bound. 12 This was recently improved to a logarithmic separation in [JMNŽ12], obtained for XORpebbling contradictions over line graphs, i.e., graphs with vertex sets {v 1 , . .…”

“…is an interesting complexity measure with nontrivial relations to proof length and width. We note that apart from being of theoretical interest, clause space has also been proposed in [ABLM08] as a relevant measure of the hardness in practice of CNF formulas for SAT solvers, and such possible connections have been further investigated in [JMNŽ12]. The resolution proof system was generalized by Krajíček [Kra01], who introduced the the family of k-DNF resolution proof systems as an intermediate step between resolution and depth-2 Frege systems.…”

Unified Characterisations of Resolution Hardness Measures

“…However, size and space are not the only measures which are interesting with respect to SAT solving, and the question what constitutes a good hardness measure for practical SAT solving is a very important one (cf. [4,38] for discussions).…”

“…The separation awid → css is shown in [51], crs → cts in [38], hd → dep and wid → dep use unsatisfiable Horn 3-clause-sets, and dep → n uses unsatisfiable clause-sets which are not minimally unsatisfiable. These measures do not just apply to unsatisfiable clause-sets, but are extended to satisfiable clause-sets, taking a worst-case approach over all unsatisfiable sub-instances obtained by applying partial assignments (instantiations).…”