The decision problem

Given a Boolean formula $\phi$, does $\phi$ have exactly one satisfying assignment?

can be seen to be in $\Delta_2$, $\mathsf{UP}$-hard and $\mathsf{coNP}$-hard. Is anything more known about its complexity?


Your problem is known as the $\text{UNIQUE-SAT}$ problem which is $\mathsf{US}$-complete. The problem is in $\mathsf{D^p}$ but not known to be $\mathsf{D^p}$-hard under deterministic polynomial time reductions, where the class $\mathsf{D^p} = \{ L_1 \cap \overline{L_2} \mid L_1,L_2 \in \mathsf{NP} \}$.

It was shown by Papadimitriou and Yannakis [1] that the set of uniquely satisfiable formulas is contained in $\mathsf{D^p}$. This follows by the definition of $\mathsf{D^p}$: let $L_1$ be SAT, and let $L_2$ be the set of formulas with $2$ or more satisfying assignments. Regarding $\mathsf{D^p}$-hardness of $\text{UNIQUE-SAT}$, Blass and Gurevich [2] gave a partial answer. For one, they showed a non-relativizing proof technique would be needed to settle the question. However, Valiant and Vazirani [3] gave a randomized polynomial time reduction from $\text{SAT}$ showing $\mathsf{D^p}$-hardness of $\text{UNIQUE-SAT}$ under randomized polynomial time reductions .

When it is known that the problem has at most one assignment or no assignments, the promise problem is called $\text{UNAMBIGUOUS-SAT}$. The Valiant–Vazirani theorem states that if there is a polynomial time algorithm for $\text{UNAMBIGUOUS-SAT}$, then $\mathsf{NP}=\mathsf{RP}$. To prove their theorem they showed that the promise problem $\text{UNAMBIGUOUS-SAT}$ is $\mathsf{NP}$-hard under randomized polynomial time reductions. A corollary that follows from the Valiant–Vazirani theorem is that $\text{UNIQUE-SAT}$ is complete for $\mathsf{D^p}$ under randomized polynomial time reductions.

[1] Papadimitriou, Christos H., and Mihalis Yannakakis. "The complexity of facets (and some facets of complexity)." Proceedings of the fourteenth annual ACM symposium on Theory of computing. ACM, 1982.

[2] Blass, Andreas, and Yuri Gurevich. "On the unique satisfiability problem." Information and Control 55.1 (1982): 80-88.

[3] Valiant, Leslie G., and Vijay V. Vazirani. "NP is as easy as detecting unique solutions." Theoretical Computer Science 47 (1986): 85-93.

| cite | improve this answer | |
  • $\begingroup$ Thanks for the answer; I also found a chapter in a book saying existence of a deterministic reduction is open. $\endgroup$ – sdcvvc Aug 23 '13 at 13:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.