A sharp threshold in proof complexity

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 M5S 1A4
Venue:
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Year:
2001

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Abstract

We give the first example of a sharp threshold in proof complexity. More precisely, we show that for any sufficiently small &egr;0 and &Dgr;2.28, random formulas consisting of (1-&egr;)n 2-clauses and &Dgr n 3-clauses, which are known to be unsatisfiable almost certainly, almost certainly require resolution and Davis-Putnam proofs of unsatisfiability of exponential size, whereas it is easily seen that random formulas with (1+&egr;)n 2-clauses (and &Dgr; n 3 clauses) have linear size proofs of unsatisfiability almost certainly.A consequence of our result also yields the first proof that typical random 3-CNF formulas at ratios below the generally accepted range of the satisfiability threshold (and thus expected to be satisfiable almost certainly) cause natural Davis-Putnam algorithms to take exponential time to find satisfying assignments.