摘要：This paper presents the following results on sets that are complete for $\NP$. \begin{enumerate} \item If there is a problem in $\NP$ that requires $\twonO$ time at almost all lengths, then every many-one NP-complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. \item If there is a problem in $\CoNP$ that cannot be solved by polynomial-size nondeterministic circuits, then every many-one complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. \item If there exist a one-way permutation that is secure against subexponential-size circuits and there is a hard tally language in $\NP \cap \CoNP$, then there is a Turing complete language for $\NP$ that is not many-one complete. \end{enumerate} Our first two results use worst-case hardness hypotheses whereas earlier work that showed similar results relied on average-case or almost-everywhere hardness assumptions. The use of average-case and worst-case hypotheses in the last result is unique as previous results obtaining the same consequence relied on almost-everywhere hardness results.
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