Computational Complexity and Phase Transitions

Phase transitions in combinatorial problems have recently been shown to be useful in locating "hard" instances of combinatorial problems. The connection between computational complexity and the existence of phase transitions has been addressed in Statistical Mechanics and Artificial Intelligence, but not studied rigorously. We take a step in this direction by investigating the existence of sharp thresholds for the class of generalized satisfiability problems defined by Schaefer. In the case when all constraints are clauses we give a complete characterization of such problems that have a sharp threshold. While NP-completeness does not imply (even in this restricted case) the existence of a sharp threshold, it "almost implies" this, since clausal generalized satisfiability problems that lack a sharp threshold are either 1. polynomial time solvable, or 2. predicted, with success probability lower bounded by some positive constant by across all the probability range, by a single, trivial procedure.Comment: A (slightly) revised version of the paper submitted to the 15th IEEE Conference on Computational Complexit

Phase Transitions and Backbones of Constraint Minimization Problems

Many real-world problems involve constraints that cannot be all satisfied. The goal toward an overconstrained problem is to find solutions minimizing the total number of constraints violated. We call such a problem constraint minimization problem (CMP). We study the behavior of the phase transitions and backbones of CMP. We first investigate the relationship between the phase transitions of Boolean satisfiability, or precisely 3-SAT (a well-studied NP-complete decision problem), and the phase transitions of MAX 3-SAT (an NP-hard optimization problem). To bridge the gap between the easy-hard-easy phase transitions of 3-SAT, in which solutions of bounded quality, e.g., solutions with at most a constant number of constraints violated, are sufficient. We show that phase transitions are persistent in bounded 3-SAT and are similar to that of 3-SAT. We then study backbones of MAX 3-SAT, which are critically constrained variables that have fixed values in all optimal solutions. Our experimental results show that backbones of MAX 3-SAT emerge abruptly and experience sharp transitions from nonexistence when underconstrained to almost complete when overconstrained. More interestingly, the phase transitions of MAX 3-SAT backbones surprisingly concur with the phase transitions of satisfiability of 3-SAT. Specifically, the backbone of MAX 3-SAT with size 0.5 approximately collocates with the 0.5 satisfiablity of 3-SAT, and hte backbone and satisfiability seems to follow a linear correlation near this 0.5-0.5 collocation

Random k-SAT and the Power of Two Choices

We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well-studied area of probabilistic combinatorics (Achlioptas processes) to random CSP's. In particular, while a rule to delay the 2-SAT threshold was known previously, this is the first proof of a rule to shift the threshold of k-SAT for k >= 3. We then propose a gap decision problem based upon this semi-random model. The aim of the problem is to investigate the hardness of the random k-SAT decision problem, as opposed to the problem of finding an assignment or certificate of unsatisfiability. Finally, we discuss connections to the study of Achlioptas random graph processes.Comment: 13 page