Phase transitions of PP-complete satisfiability problems

  • Authors:

  • Delbert D. Bailey;Víctor Dalmau;Phokion G. Kolaitis

  • Affiliations:

  • Computer Science Department, University of California, Santa Cruz, Santa Cruz, CA 95064, USA;Departament de tecnologia, Universitat Pompeu Fabra,Estació de França, Passeig de la circumval.lació 8, Barcelona 08003, Spain;Computer Science Department, University of California, Santa Cruz, Santa Cruz, CA 95064, USA

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2007

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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.