Phase Transition and Network Structure in Realistic SAT Problems

Abstract

Previous research has shown that 3-SAT problems are easy to solve both when the “constrainedness” (the ratio of the number of clauses to the number of variables) is low and when it is high, abruptly transitioning from easy to hard in a very narrow region of constrainedness. Most of these “phase transition” studies were done on SAT instances that follow uniform random distribution. In such a distribution, variables take part in clauses with uniform probability, and clauses are independent (uncorrelated). The assumptions of uniform random distribution are, however, not satisfied when we consider SAT instances that result from real problems. Our project aims for a deeper understanding of the hardness of SAT problems that arise in practice. In particular, we study two key questions: (1) How does the phase transition behavior change with more realistic and natural distributions of SAT problems? and (2) Can we gain an understanding of the phase transition in terms of the network structure of these SAT problems? Our hypothesis is that the network properties help predict and explain how the easy-to-hard problem transition for realistic SAT problems differs from those for uniform random distribution.