Phase Transitions of Bounded Satisfiability Problems

Abstract

We investigate phase transitions for the family of bounded satisfiability problems 3SAT(b), recently introduced by Zhang, that ask: given a 3CNF- formula, is there a truth assignment that violates no more than b of its clauses. Zhang's results were experimental and for a fixed number of variables (n = 25), and suggested that the locations of the phase transitions for 3SAT(b) are separated and move significantly as b increases. Analysis of these locations was posed as an open question. We analytically show that the phase transitions of all 3SAT(6) problems must occur within a narrow region, regardless of how large the value of b is. Moreover, our experiments reveal that the phase transitions for these problems occur in a remarkable way. Specifically, unlike 3SAT, the probability curves for 3SAT(6) do not have a quasi-common intersection point about which they rotate as they become steeper with increasing n. Instead, they move rapidly to the left toward the narrow region that the analysis predicts.