Abstract
In this paper, we measure “intractability” of complexity classes by considering polynomial time 1-truth-table reducibility (in short, ≤ 1−tt P-reducibility) to a sparse set. We mainly investigate nondeterministic complexity classes that are defined in relation to one-way functions: UP, FewP, UBPP, and UP . We show that if UP (resp., UBPP and UP . has a polynomial time unsolvable problem, then it indeed has a problem that is “tractable”ot only by being polynomial time unsolvable, but also by being ≤ 1−tt P-reducible to no sparse set. As an immediate consequence of our observation, we can also prove that if R ≠ NP (resp., P ≠ FewP and P ≠ UP ) then no NP-complete set is ≤ 1−tt P-reducible to a sparse set, and thus no NP-complete set has a p-close approximation; this provides a partial answer to a question asked by Schöning.
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