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This paper presents the following results on sets that are complete for NP.  1. If there is a problem in NP that requires exponential time at almost alllengths, then every many-one NP-complete set is complete underlength-increasing reductions that are computed by polynomial-size circuits. 2.If there is a problem in coNP that cannot be solved by polynomial-sizenondeterministic circuits, then every many-one complete set is complete underlength-increasing reductions that are computed by polynomial-size circuits. 3.If there exist a one-way permutation that is secure against subexponential-sizecircuits and there is a hard tally language in NP intersect coNP, then there isa Turing complete language for NP that is not many-one complete. Our first tworesults use worst-case hardness hypotheses whereas earlier work that showedsimilar results relied on average-case or almost-everywhere hardnessassumptions. The use of average-case and worst-case hypotheses in the lastresult is unique as previous results obtaining the same consequence relied onalmost-everywhere hardness results.

Collapsing and Separating Completeness Notions under Average-Case and Worst-Case Hypotheses