Adversarial Evolution: Phase transition in non-uniform hard satisfiability problems

Abstract

What makes a combinatorial optimization problem hard? The concept of phase transition was introduced in combinatorial decision problems to explain that not all NP-Complete problems are hard, and that there exists a phase transition from solvable to unsolvable problems, within which hard problems exist. Phase transition has been studied using randomly generated problems in which variables have uniform distributions across the different constraints. Real-world problems demonstrate different distributions, however. This paper reveals the relationship between the difficulty of a 3-SAT problem and graph properties. It establishes for the first time a link between the theory of phase-transition in 3-SAT and phase transitions in complex systems and networks. This paper also addresses the question of whether the phase transition phenomenon exists for non-uniform randomly generated problems. A positive answer to this question means in principle that (1) we can generate test problems for combinatorial optimization that are not uniform; (2) we can generate test problems that resemble hard versions of real-world problems; (3) we can identify the features that we need to look for in a problem to test whether or not it is hard. We use a method that we call Adversarial Evolution (AE). In AE, an evolutionary computation method is used to generate hard problem instances by evolving solutions towards the failure of an algorithm and the phase transition region.