The Satisfiability Threshold for k-XORSAT

TL;DR

This paper establishes the sharp threshold for satisfiability in random k-XORSAT problems for all k ≥ 3, extending previous results and analyzing phase transition behavior in constrained and unconstrained models.

Contribution

It proves that the satisfiability threshold at m/n=1 holds for all k ≥ 3 in constrained k-XORSAT and extends this to unconstrained models using hypergraph core analysis.

Findings

Threshold at m/n=1 for all k ≥ 3 in constrained k-XORSAT.

Sharp phase transition window for constrained k-XORSAT.

Extension of threshold results to unconstrained k-XORSAT models.

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.