The Boolean satisfiability problem (SAT) is a fundamental NP-complete decision problem in automated reasoning and mathematical logic. As evidenced by the results of SAT competitions, the performance of SAT solvers varies substantially between different SAT categories (random, crafted, and industrial). A suggested explanation is that SAT solvers may exploit the underlying structure inherent to SAT instances. There have been attempts to define the structure of SAT in terms of structural measures such as phase transition, backbones, backdoors, small-world, scale-free, treewidth, centrality, community, self-similarity, and entropy. Still, the empirical evidence of structural measures for SAT has been provided for only some SAT categories. Furthermore, the evidence has not been theoretically proven. Also, the impact of structural measures on the behavior of SAT solvers has not been extensively examined. This work provides a comprehensive study on structural measures for SAT that have been presented in the literature. We provide an overview of the works on structural measures for SAT and their relatedness to the performance of SAT solvers. Accordingly, a taxonomy of structural measures for SAT is presented. We also review in detail important applications of structural measures for SAT, focusing mainly on enhancing SAT solvers, generating SAT instances, and classifying SAT instances.

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Boolean structural measures were introduced to explain the high performance of conflict-driven clause-learning (CDCL) SAT solvers on industrial SAT instances. Those considered in this study include measures related to backbones and backdoors: backbone size, backbone frequency, and backdoor size. A key area of research is to improve the performance of CDCL SAT solvers by exploiting these measures. For the purpose of guiding the CDCL SAT solver for branching on backbone and backdoor variables, this study proposes low-overhead heuristics for computing these variables. Through these heuristics, a set of modifications to the Variable State Independent Decaying Sum (VSIDS) decision heuristic is suggested to exploit backbones and backdoors and potentially improve the performance of CDCL SAT solvers. In total, fifteen variants of two competitive base solvers, MapleLCMDistChronoBT-DL-v3 and LSTech, were developed. Empirical evaluation was conducted on 32 industrial families from 2002–2021 SAT competitions. According to the results, modifying the VSIDS heuristic in the base solvers to exploit backbones and backdoors improves its performance. In particular, our new CDCL SAT solver, LSTech_BBsfcr_v1, solved more industrial SAT instances than the winning CDCL SAT solvers in 2020 and 2021 SAT competitions.