A sharp threshold in proof complexity yields lower bounds for satisfiability search

Authors:
Dimitris Achlioptas;Paul Beame;Michael Molloy
Affiliations:
Microsoft Research, One Microsoft Way, Redmond, WA;Computer Science and Engineering, University of Washington, Seattle, WA;Department of Computer Science, University of Toronto, Toronto, Ontario, Canada M5S 1A4
Venue:
Journal of Computer and System Sciences - STOC 2001
Year:
2004

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Abstract

Let F(ρn, Δn) denote a random CNF formula consisting of ρn randomly chosen 2-clauses and Δn randomly chosen 3-clauses, for some arbitrary constants ρ, Δ ≥ 0. It is well-known that, with probability 1 - o(1), if ρ 1 then F(ρn, Δn) has a linear-size resolution refutation. We prove that, with probability 1 - o(1), if ρ F(ρn, Δn) has no subexponential-size resolution refutation.Our result also yields the first proof that random 3-CNF formulas with densities well below the generally accepted range of the satisfiability threshold (and thus believed to be satisfiable) cause natural Davis-Putnam algorithms to take exponential time (to find a satisfying assignment).