The Satisfiability Threshold for $k$-XORSAT, using an alternative proof

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

This version integrates the previous version's alternative proof into the paper (see arXiv:1212.1905). Other proofs are amended, and the paper's structure is clarified. The main result is improved: the phase transition occurs for an arbitrarily slowly growing gap between m and n

References in corpus (3)

• The Satisfiability Threshold for k-XORSAT
• Poisson Cloning Model for Random Graphs
• Rank deficiency in sparse random GF[2] matrices

Cited by in corpus (1)

• Concentration of the number of solutions of random planted CSPs and Goldreich's one-way candidates