On intractability of the classUP

The classUP V] is the class of sets accepted by polynomial-time nondeterministic Turing machines which have at most one accepting path for every input. The complexity of this class closely relates to that of computing inverses ofone-way functions, where a one-way function is a one-to-one, length-increasing, and polynomial-time computable function whose inverse cannot be computed within polynomial time. It is known GS], K] that there exists a one-way function if and only ifP UP. In this paper the intractability of sets inUP is investigated in terms of polynomial-time reducibility to a sparse set. It is shown thatUP has a set that is _m ^P -reducible to no sparse set ifP UP. We interpret this structural property in the relation with approximation algorithms: it is shown that ifP UP, thenUP has a set with no 1-APT approximation and, furthermore,UP has a set that is not _m ^P -reducible to any set with a 1-APT approximation. The implication of this result in the study of one-way functions is also discussed. In order to prove the main theorem, we introduce a variation of tree-pruning methods.This paper was written while the author visited the Department of Mathematics, University of California, Santa Barbara. This research was supported in part by the National Science Foundation under Grant CCR-8611980.