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On the Random Satisfiable Process

Published online by Cambridge University Press:  01 September 2009

MICHAEL KRIVELEVICH ,
BENNY SUDAKOV  and
DAN VILENCHIK
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel (e-mail: krivelev@post.tau.ac.il)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, UCLA, Los Angeles, USA (e-mail: bsudakov@math.ucla.edu)
DAN VILENCHIK
Affiliation:
Department of Mathematics, UCLA, Los Angeles, USA (e-mail: danny.vilenchik@gmail.com)
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Abstract

In this work we suggest a new model for generating random satisfiable k-CNF formulas. To generate such formulas. randomly permute all possible clauses over the variables x1,. . .,xn, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if, after its addition, the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first m clauses (in the random permutation's order).

Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruciński and Wormald in 1992 for graphs with a fixed degree sequence, and also by Erdős, Suen and Winkler in 1995 for triangle-free and bipartite graphs. Since then many other graph properties have been studied, such as planarity and H-freeness. Thus our model is a natural extension of this approach to the satisfiability setting.

Our main contribution is as follows. For m ≥ cn, c = c(k) a sufficiently large constant, we are able to characterize the structure of the solution space of a typical formula in this distribution. Specifically, we show that typically all satisfying assignments are essentially clustered in one cluster, and all but e−Ω(m/n)n of the variables take the same value in all satisfying assignments. We also describe a polynomial-time algorithm that finds w.h.p. a satisfying assignment for such formulas.

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 5: New Directions in Algorithms, Combinatorics and Optimization , September 2009 , pp. 775 - 801
Copyright
Copyright © Cambridge University Press 2009

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