Pittel, Boris and Sorkin, Gregory B.
(2015)
The satisfiability threshold for kXORSAT.
Combinatorics, Probability and Computing, 25 (2).
pp. 236268.
ISSN 09635483
Abstract
We consider "unconstrained" random $k$XORSAT, which is a uniformly random system of $m$ linear nonhomogeneous 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 3XORSAT, and analyzed the 2core of a random 3uniform hypergraph to extend this result to find the threshold for unconstrained 3XORSAT. 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 2core 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 $mn \to \infty$ implies almostsure satisfiability, while $mn \to +\infty$ implies almostsure unsatisfiability.
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