Uniquely satisfiable k-SAT instances with almost minimal occurrences of each variable

• Authors: William Matthews;Ramamohan Paturi
• Venue: SAT'10 Proceedings of the 13th international conference on Theory and Applications of Satisfiability Testing
• Year: 2010

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. Kratochvíl, 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 sf(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.