The phase transition in random regular exact cover

Abstract

A k-uniform, d-regular instance of Exact Cover is a family of m sets {Sj ⊆ (1,…,n}, where each subset has size k and each 1≤ i ≤ n is contained in d of the Sj. It is satisfiable if there is a subset T ⊆ {1,…,n} such that |T ∩ Sj| = 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 (Formula presented) we show that Fn,d,k is satisfiable with high probability if d < d* and unsatisfiable with high probability if d > d*. We do this with a simple application of the first and second moment methods, boosting the probability of satisfiability below d* to 1 – o(1) using the small subgraph conditioning method.