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 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. © 2010 Springer-Verlag.
CITATION STYLE
Matthews, W., & Paturi, R. (2010). Uniquely satisfiable k-SAT instances with almost minimal occurrences of each variable. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6175 LNCS, pp. 369–374). https://doi.org/10.1007/978-3-642-14186-7_34
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