Phase Transitions of PP-Complete Satisfiability Problems

Delbert D. Bailey, Víctor Dalmau, Phokion G. Kolaitis

Abstract: The complexity class PP consists of all decision problems solvable by polynomial-time probabilistic Turing machines. It is well known that PP is a highly intractable complexity class and that PP-complete problems are in all likelihood harder than NP-complete problems. We investigate the existence of phase transitions for a family of PP-complete Boolean satisfiability problems under the fixed clauses-to-variables ratio model. A typical member of this family is the decision problem # 3SAT(>=2^n^/^2): given a 3CNF-formula, is it satisfied by at least the square-root of the total number of possible truth assignments? We provide evidence to the effect that there is a critical ratio r"3","2 at which the asymptotic probability of # 3SAT(>=2^n^/^2) undergoes a phase transition from 1 to 0. We obtain upper and lower bounds for r"3","2 by showing that 0.9227==2^n^/^2) using a natural modification of the Davis-Putnam-Logemann-Loveland (DPLL) procedure. Our experimental results suggest that r"3","2~2.5. Moreover, the average number of recursive calls of this modified DPLL procedure reaches a peak around 2.5 as well.