Decomposing Hard SAT Instances with Metaheuristic Optimization

In the article, within the framework of the Boolean Satisfiability problem (SAT), the problem of estimating the hardness of specific Boolean formulas w.r.t. a specific complete SAT solving algorithm is considered. Based on the well-known Strong Backdoor Set (SBS) concept, we introduce the notion of decomposition hardness (d-hardness). If \(B\) is an arbitrary subset of the set of variables occurring in a SAT formula \(C\), and \(A\) is an arbitrary complete SAT solver , then the d-hardness expresses an estimate of the hardness of \(C\) w.r.t. \(A\) and \(B\). We show that the d-hardness of \(C\) w.r.t. a particular \(B\) can be expressed in terms of the expected value of a special random variable associated with \(A\), \(B\), and \(C\). For its computational evaluation, algorithms based on the Monte Carlo method can be used. The problem of finding \(B\) with the minimum value of d-hardness is formulated as an optimization problem for a pseudo-Boolean function whose values are calculated as a result of a probabilistic experiment. To minimize this function, we use evolutionary algorithms. In the experimental part, we demonstrate the applicability of the concept of d-hardness and the methods of its estimation to solving hard unsatisfiable SAT instances.