# The satisfiability threshold for k-XORSAT

Pittel, Boris and Sorkin, Gregory B. (2015) The satisfiability threshold for k-XORSAT. Combinatorics, Probability and Computing, 25 (2). pp. 236-268. ISSN 0963-5483

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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 $k \geq 3$ 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 analyzed 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 $k\ge 3$, 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 $m-n \to -\infty$ implies almost-sure satisfiability, while $m-n \to +\infty$ implies almost-sure unsatisfiability.

Item Type: Article http://journals.cambridge.org/action/displayJourna... © 2015 Cambridge University Press LSE Q Science > QA Mathematics Research centres and groups > Operations Research Group 06 May 2015 08:29 20 Jul 2019 02:00 http://eprints.lse.ac.uk/id/eprint/61798