NP is as easy as detecting unique solutions

Authors:
L G Valiant;V V Vazirani
Affiliations:
Aiken Computation Laboratory, Harvard University;Computer Science Department, Cornell University
Venue:
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Year:
1985

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Abstract

For all known NP-complete problems the number of solutions in instances having solutions may vary over an exponentially large range. Furthermore, most of the well-known ones, such as satisfiability, are parsimoniously interreducible, and these can have any number of solutions between zero and an exponentially large number. It is natural to ask whether the inherent intractability of NP-complete problems is caused by this wide variation. In this paper we give a negative answer to this using randomized reductions. We show that the problems of distinguishing between instances of SAT having zero or one solution, or finding solutions to instances of SAT having unique solutions, are as hard as SAT itself. Several corollaries about the difficulty of specific problems follow. For example if the parity of the number of solutions of SAT can be computed in RP then NP = RP. Some further problems can be shown to be hard for NP or DP via randomized reductions.