On the Unique Satisfiability Problem

Satisfiability Parsimoniously Reduces to the TantrixTM Rotation Puzzle Problem

Holzer and Holzer [HH04] proved that the Tantrix rotation puzzle problem is NP-complete. They also showed that for infinite rotation puzzles, this problem becomes undecidable. We study the counting version and the unique version of this problem. We prove that the satisfiability problem parsimoniously reduces to the Tantrix rotation puzzle problem. In particular, this reduction preserves the uni...

Unique Solution Instance Generation for the 3-Satisfiability (3SAT) Problem

The Complexity of Some Subclasses of Minimal Unsatis able Formulas

This paper is concerned with the complexity of some natural subclasses of minimal unsatisfiable formulas. We show the D –completeness of the classes of maximal and marginal minimal unsatisfiable formulas. Then we consider the class Unique–MU of minimal unsatisfiable formulas which have after removing a clause exactly one satisfying truth assignment. We show that Unique–MU has the same complexit...

Unique k-SAT is as Hard as k-SAT

In this work we show that Unique k-SAT is as hard as k-SAT for every k ∈ N. This settles a conjecture by Calabro, Impagliazzo, Kabanets and Paturi [4]. To provide an affirmative answer to this conjecture, we develop a randomness optimal construction of Isolation Lemma for k-SAT.

The Complexity of Planar Counting Problems

We prove the #P-hardness of the counting problems associated with various satisfiability, graph, and combinatorial problems, when restricted to planar instances. These problems include 3Sat, 1-3Sat, 1-Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into Triangles, and Clique Cover. We also prove the NP-completeness of the Ambiguous Satisfiabilit...