On the Structure of the Boolean Satisfiability Problem: A Survey

Abstract

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.

Boosting the Performance of CDCL-Based SAT Solvers by Exploiting Backbones and Backdoors

“…Boolean structure measures well known in the literature are: phase transition, backbone size, backdoor size, and backbone/backdoor variable overlap size [25]. In view of the fact that the phase transition is only located for random and industrial-like instances, we focus on characterizing SAT instances based on backbones and backdoors.…”

“…When the solver reaches a promising partial assignment, it enters a non-backtracking stage until it obtains a full assignment (line 34). The SLS solver, CCAr, is then called every number of restarts (sthreshold) in order to find a satisfying assignment close to the full assignment (lines [16][17][18][19][20][21][22][23][24][25][26][27][28]. This value of sthreshold is set at 300, but if the SLS solution does not improve, it increases; otherwise, it decreases, keeping the value above 300 (lines 24-27).…”

“…Several studies have shown, for example, that backbone/backdoor-guided branching heuristics improve the solver performance [18,[22][23][24]. Despite these efforts, this problem remains an open issue [19,25]. One reason is that the computational estimation of most Boolean structural measures (e.g., backbone and backdoor) is intractable [26].…”

“…One reason is that the computational estimation of most Boolean structural measures (e.g., backbone and backdoor) is intractable [26]. Second, as a consequence, the structural measures of SAT and their impact on the behavior of SAT solvers have not been fully investigated [19,25].…”

Reasoning and inference for (Maximum) satisfiability: new insights

“…However, modern solvers are extremely efficient and manage to outstandingly solve formulas with millions of variables and clauses. Many studies have attempted to investigate this conflicting gap between the theoretical difficulty of SAT and the remarkable practical performance of modern solvers [AMM22a].…”