Uniquely Satisfiable k-SAT Instances with Almost Minimal Occurrences of Each Variable

Let (k, s)-SAT refer the family of satisfiability problems restricted to CNF formulas with exactly k distinct literals per clause and at most s occurrences of each variable. Kratochv́ıl, Savický and Tuza [6] show that there exists a function f(k) such that for all s ≤ f(k), all (k, s)-SAT instances are satisfiable whereas for k ≥ 3 and s > f(k), (k, s)-SAT is NP-complete. We define a new function u(k) as the minimum s such that uniquely satisfiable (k, s)-SAT formulas exist. We show that for k ≥ 3, unique solutions and NP-hardness occur at almost the same value of s: f(k) ≤ u(k) ≤ f(k) + 2. We also give a parsimonious reduction from SAT to (k, s)-SAT for any k ≥ 3 and s ≥ f(k) + 2. When combined with the Valiant–Vazirani Theorem [8], this gives a randomized polynomial time reduction from SAT to UNIQUE-(k, s)-SAT.