Phase Transition for Maximum Not-All-Equal Satisfiability

Phase transition is a dramatic transition from one state to another state when a particular parameter varies. This paper aims to study the phase transition of maximum not-all-equal satisfiability problem (Max NAE SAT), an optimization of not-all-equal satisfiability problem (NAE SAT). Given a conjunctive normal formula (CNF) F with n variables and rn k-clauses (the clause exactly contains k literals), we use first-moment method to obtain an upper bound for f (n, rn) the expectation of the maximum number of NAE-satisfied clauses of random Max NAE k-SAT. In addition, we also consider the phase transition of decision version of random Max NAE k-SAT bounded not-all-equal satisfiability problem (NAE k-SAT(b)). We demonstrate that there is a phase transition point rk,b separating the region where almost all NAE k-SAT(b) instances can be solved from the region where almost all NAE k-SAT(b) instances can't be solved. Furthermore, we analyze the upper bound and lower bound for r(k,b).