Investigating Computational Hardness through Realistic Boolean SAT Distributions

A fundamental question in Computer Science is understanding when a specific class of problems go from being computationally easy to hard. Because of its generality and applications, the problem of Boolean Satisfiability (aka SAT) is often used as a vehicle for these studies. A signal result from these studies is that the hardness of SAT problems exhibits a dramatic easy-to-hard phase transition with respect to the problem constrainedness. Past studies have however focused mostly on SAT instances generated using uniform random distributions, where all constraints are independently generated, and the problem variables are all considered of equal importance. These assumptions are unfortunately not satisfied by most real problem. Our project aims for a deeper understanding of hardness of SAT problems that arise in practice. We study two key questions: (i) How does easy-to-hard transition change with more realistic distributions that capture neighborhood sensitivity and rich-get-richer aspects of real problems and (ii) Whether the results on random SAT distributions shed light on the hardness of specific problems such as Sudoku. Our results, based on extensive empirical studies with our own instrumented SAT solver, provide important insights into the practical implications of phase transition behavior on SAT problems. Executive Summary: In this project, we aimed to understand how real-world computational tasks go from being easy to being hard in terms of the time taken to solve them. We used the problem of Boolean Satisfiability (SAT) as the focus of our study because many other problems can be translated into SAT problems. SAT problems involve assigning values to a set of Boolean (True/False) decision variables while satisfying all the given constraints. They have applications ranging from scheduling software to decision making for robots. Prior research has shown that random SAT problems exhibit a dramatic easy-to-hard transition in problem hardness as their constrainedness (number of constraints relative to number of decision variables) is varied. The main focus past work has however been on uniform random distributions, in which clauses are generated independently of other clauses. Our project aimed for a deeper understanding of hardness of SAT problems that arise in practice. We studied two key questions: (i) How does easy-to-hard transition change with more realistic distributions that capture neighborhood sensitivity and rich-get-richer aspects of real problems and (ii) Whether the results on random SAT distributions shed light on the hardness of specific problems such as Sudoku. Our results, backed by extensive empirical studies with …