The impact of heterogeneity and geometry on the proof complexity of random satisfiability

Abstract Satisfiability is considered the canonical NP‐complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large‐scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real‐world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random ‐SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random ‐SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT‐solvers. In contrast, modeling locality with underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time.