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The Satisfiability Threshold for k-XORSAT

Part of: Theory of computing Probability theory on algebraic and topological structures Combinatorics

Published online by Cambridge University Press:  31 July 2015

BORIS PITTEL  and
GREGORY B. SORKIN
BORIS PITTEL
Affiliation:
Department of Mathematics, Ohio State University, Columbus OH 43210, USA (e-mail: [email protected])
GREGORY B. SORKIN
Affiliation:
Departments of Management and Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK (e-mail: [email protected])
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Abstract

We consider ‘unconstrained’ random k-XORSAT, which is a uniformly random system of m linear non-homogeneous equations in $\mathbb{F}$2 over n variables, each equation containing k ⩾ 3 variables, and also consider a ‘constrained’ model where every variable appears in at least two equations. Dubois and Mandler proved that m/n = 1 is a sharp threshold for satisfiability of constrained 3-XORSAT, and analysed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT.

We show that m/n = 1 remains a sharp threshold for satisfiability of constrained k-XORSAT for every k ⩾ 3, and we use standard results on the 2-core of a random k-uniform hypergraph to extend this result to find the threshold for unconstrained k-XORSAT. For constrained k-XORSAT we narrow the phase transition window, showing that m − n → −∞ implies almost-sure satisfiability, while m − n → +∞ implies almost-sure unsatisfiability.

MSC classification

Primary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see ) 05A16: Asymptotic enumeration
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 2 , March 2016 , pp. 236 - 268
Copyright
Copyright © Cambridge University Press 2015 

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