Popularity-similarity random SAT formulas

Abstract

In the last decades, we have witnessed a remarkable success of algorithms solving the Boolean Satisfiability problem (SAT) on instances encoding application or real-world problems arising from a very diverse number of domains, such as hardware and software verification, planning or cryptography. These algorithms are the so known Conflict-Driven Clause Learning (CDCL) SAT solvers. Interestingly enough, the reasons for the success of these solvers on this diverse range of problems are not completely understood yet.A common issue when facing this open challenge is the heterogeneity of this set of benchmarks. Another problem is the limited number of existing instances. In this context, random models of SAT formulas capturing features shared by the majority of these application benchmarks become crucial, for both theoretical and practical purposes. On the one hand, it is undoubtedly necessary to have random models where theoretical properties, like hardness, can be studied. Therefore, realistic random SAT models may contribute to explain the success of these solvers on these industrial problems. On the other hand, the limited number of benchmarks and their hardness in practice makes the evaluation of new solving techniques a costly task. Therefore, these realistic random SAT generators can provide an unlimited number of pseudo-industrial random SAT instances with some desired properties.In this work, we present a random SAT instances generator based on the notion of locality. This notion is complementary to the popularity of variables, which is present in the scale-free structure, observable in actual application problems and achievable by previous generators. Our random SAT model combines both locality and popularity, and we show that they are two decisive dimensions of attractiveness among the variables of a formula, and how CDCL SAT solvers take advantage of them. Locality is closely related to the community structure, another important feature of application SAT benchmarks, which is indirectly achieved by this model. To the best of our knowledge, this is the first random SAT model that generates both scale-free structure and community structure at once.