Rigorous results for random (2 + p)-SAT

  • Authors:

  • Dimitris Achlioptas;Lefteris M. Kirousis;Evangelos Kranakis;Danny Krizanc

  • Affiliations:

  • Microsoft Research, Redmond, WA;University of Patras, Patras, Greece;Carleton University, Ottawa, Ontario, Canada;Carleton University, Ottawa, Ontario, Canada

  • Venue:
  • Theoretical Computer Science - Phase transitions in combinatorial problems
  • Year:
  • 2001

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Abstract

In recent years there has been significant interest in the study of random k-SAT formulae. For a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct, non-complementary literals from its variables (k-clauses). A random k-SAT formula Fk (n;m) is formed by selectinguniformly and independently m clauses from Bk and takingtheir conjunction. Motivated by insights from statistical mechanics that suggest a possible relationship between the "order" of phase transitions and computational complexity, Monasson and Zecchina (Phys. Rev. E 56(2) (1997) 1357) proposed the random (2+p)-SAT model: for a given p . [0; 1], a random (2 + p)-SAT formula, F2+p(n;m), has m randomly chosen clauses over n variables, where pm clauses are chosen from B3 and (1 - p)m from B2. Usingthe heuristic "replica method" of statistical mechanics, Monasson and Zecchina gave a number of non-rigorous predictions on the behavior of random (2 + p)-SAT formulae. In this paper we give the 1rst rigorous results for random (2 + p)-SAT, including the following surprising fact: for p ≤ 2/5, with probability 1 - o(1), a random (2 + p)-SAT formula is satisfiable if its 2-SAT subformula is satisfiable. That is, for p 6 2=5, random (2 + p)-SAT behaves like random 2-SAT.