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Branching Process Approach for 2-Sat Thresholds

Part of: Decision theory Equilibrium statistical mechanics

Published online by Cambridge University Press:  14 July 2016

Elchanan Mossel  and
Arnab Sen
Elchanan Mossel*
Affiliation:
University of California, Berkeley
Arnab Sen*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Statistics, University of California, Berkeley, CA 94720-3860, USA.
∗ Postal address: Department of Statistics, University of California, Berkeley, CA 94720-3860, USA.
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Abstract

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It is well known that, as n tends to ∞, the probability of satisfiability for a random 2-SAT formula on n variables, where each clause occurs independently with probability α / 2n, exhibits a sharp threshold at α = 1. We study a more general 2-SAT model in which each clause occurs independently but with probability αi / 2n, where i ∈ {0, 1, 2} is the number of positive literals in that clause. We generalize the branching process arguments used by Verhoeven (1999) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.

Keywords

2-SATsatisfiabilityphase transitiontwo-type branching process

MSC classification

Secondary: 62C05: General considerations 82B26: Phase transitions (general)

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 796 - 810
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Supported by NSF grants DMS 0528488 and DMS 0548249, and ONR grant N0014-07-1-05-06.

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