The Satisfiability Threshold for K-XOR Games

TL;DR

This paper establishes that the threshold ratio of equations to variables for K-XOR games matches that of K-XORSAT, confirming a phase transition point for these structured constraint satisfaction problems.

Contribution

It proves the existence and exact value of the satisfiability threshold for K-XOR games, linking it to the well-studied K-XORSAT problem.

Findings

The satisfiability threshold for K-XOR games exists.

It equals the threshold for K-XORSAT.

The threshold is characterized by the ratio m/n.

Abstract

A $K$-XORGAME system corresponds to a $K$-XORSAT system with the additional restriction that the variables divide uniformly into $K$ blocks. This forms a system of $m$ equations with $K n$ unknowns over $\mathbb{Z}_2$, and a perfect strategy corresponds to a solution to these equations. Equivalently, such equations correspond to colorings of a $K$-uniform $K$-partite hypergraph. This paper proves that the satisfiability threshold of $m/n$ for $K$-XORGAME problems exists and equals the satisfiability threshold for $K$-XORSAT.