The phase transition in random regular exact cover

Abstract

A $k$-uniform, $d$-regular instance of EXACT COVER is a family of $m$ sets $F_{n,d,k}=\{S_j\subseteq \{1,\dots ,n\}\}$, where each subset has size $k$ and each $1\le i\le n$ is contained in $d$ of the $S_j$. It is satisfiable if there is a subset $T\subseteq \{1,\dots ,n\}$ such that $|T\cap S_j|=1$ for all $j$. Alternately, we can consider it a $d$-regular instance of POSITIVE 1-IN-$k$ SAT, i.e., a Boolean formula with $m$ clauses and $n$ variables where each clause contains $k$ variables and demands that exactly one of them is true. We determine the satisfiability threshold for random instances of this type with $k>2$. Letting

we show that $F_{n,d,k}$ is satisfiable with high probability if $d<d^{\star }$ and unsatisfiable with high probability if $d>d^{\star }$. We do this with a simple application of the first and second moment methods, boosting the probability of satisfiability below $d^{\star }$ to $1-o(1)$ using the small subgraph conditioning method.