On Unique Satisfiability and Randomized Reductions

Ever since Valiant and Vazirani [VV86] showed that there exists a random reduction from SAT to USAT, the complexity of USAT has been cited as “USAT is complete for DP under randomized reductions.” However, the definition of the randomized reduction was never quite satisfying because the probability of a “correct” reduction can approach zero as the length of the formula increases. The discrepancy between the Valiant-Vazirani definition and the earlier Adleman-Manders [AM77] definition has been noted previously [Joh85]. This column reflects on recent results about the complexity of USAT and of DP which shed a new light on the meaning of completeness under randomized reductions. For example, it is pointed out that, under randomized reductions, USAT is complete for PSAT[log n] as well. These results also show that the non-robustness of DP creates many difficulties in defining a randomized reduction which gives a meaningful notion of completeness. 1 An Historical Account The central question in structural complexity theory is of course the P =? NP question. One way to consider this problem is to investigate the complexity of the NP-complete set SAT. In addition to studying the complexity of detecting a satisfiable Boolean formula, much research has been devoted to studying the complexity of computing the actual number of satisfying assignments. As a special case, the unique satisfiability problem is the problem of detecting if a Boolean formula has exactly one satisfying assignment. From the beginning, the study of the complexity of unique satisfiability has been tied to the class D and to randomized reductions. Papadimitriou and Yannakakis [PY84] first defined D to study the complexity of the facets of polytopes. This research was supported in part by NSF Research Grant CCR 88-23053. Current Address: Department of Computer Science, University of Maryland, Baltimore County Campus, Baltimore, MD 21228, USA.