Uniquely Satisfiable k-SAT Instances with Almost Minimal Occurrences of Each Variable
Part of the
Lecture Notes in Computer Science
book series (LNCS, volume 6175)
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  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 , this gives a randomized polynomial time reduction from SAT to UNIQUE-(k,s)-SAT.
KeywordsConjunctive Normal Form Satisfying Assignment Polynomial Time Reduction Fresh Variable Minimal Occurrence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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