# Annales de l’Institut Henri Poincaré D

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**Volume 3, Issue 3, 2016, pp. 349–362**

**DOI: 10.4171/AIHPD/31**

Published online: 2016-09-14

The phase transition in random regular exact cover

Cristopher Moore^{[1]}(1) Santa Fe Institute, USA

A $k$-uniform, $d$-regular instance of EXACT COVER is a family of $m$ sets $F_{n,d,k} = \{ S_j \subseteq \{1,\ldots,n\} \}$, 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 $T \subseteq \{1,\ldots,n\}$ 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 \[ d^\star = \frac{\ln k}{(k-1)(- \ln (1-1/k))} + 1 \, , \] 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.

*Keywords: *Random structures, phase transitions, Boolean formulas, satisfiability, NP-complete problems, second moment method, small subgraph conditioning.

Moore Cristopher: The phase transition in random regular exact cover. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 3 (2016), 349-362. doi: 10.4171/AIHPD/31