On intractability of the classUP

Abstract

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 ≤mP-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 ≤mP-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.