The Satisfiability Threshold for k-XORSAT

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 $m-n\to -\infty$ implies almost-sure satisfiability, while $m-n\to +\infty$ implies almost-sure unsatisfiability.

Version 2 adds sharper phase transition result, new citation in literature survey, and improvements in presentation; removes Appendix treating k=3

References in corpus (2)

• Poisson Cloning Model for Random Graphs
• Rank deficiency in sparse random GF[2] matrices

Cited by in corpus (8)

• The asymptotic $k$-SAT threshold
• Catching the k-NAESAT Threshold
• The random 2-SAT partition function
• Combining the $k$-CNF and XOR Phase-Transitions
• The Satisfiability Threshold for $k$-XORSAT, using an alternative proof
• Inside the clustering window for random linear equations
• Inside the clustering threshold for random linear equations
• Concentration of the number of solutions of random planted CSPs and Goldreich's one-way candidates