Delaying satisfiability for random 2SAT

Alistair Sinclair, Dan Vilenchik

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Let (C1,C′(*)),(C2,C′(*)),...,(C m,C′(*)) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4 \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath} \pagestyle{empty} \begin{document} \begin*\binom{n}{2}\end* \end{document} clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on-line, without backtracking, but may depend on the clauses chosen previously. We show that there exists an on-line choice algorithm in the above process which results whp in a satisfiable 2CNF formula as long as m/n ≤ (1000/999)1/4. This contrasts with the well-known fact that a random m -clause formula constructed without the choice of two clauses at each step is unsatisfiable whp whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as "Achlioptas processes." This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on-line setting above, we also consider an off-line version in which all m clause-pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off-line setting, we show that the two-choice satisfiability threshold for k -SAT for any fixed k coincides with the standard satisfiability threshold for random 2k -SAT.

Original languageEnglish
Pages (from-to)251-263
Number of pages13
JournalRandom Structures and Algorithms
Volume43
Issue number2
DOIs
StatePublished - 1 Sep 2013
Externally publishedYes

Keywords

  • Achlioptas process
  • Random SAT
  • Random structures
  • Threshold Phenomenon

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