Collapsing and Separating Completeness Notions Under Average-Case and Worst-Case Hypotheses
This paper presents the following results on sets that are complete for NP. (i) If there is a problem in NP that requires \(2^{n^{\Omega(1)}}\) 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.  (ii) If there is a problem in co-NP that cannot be solved by polynomial-size nondeterministic circuits, then every many-one NP-complete set is complete under length-increasing reductions that are computed by polynomial-size circuits.  (iii) If there exist a one-way permutation that is secure against subexponential-size circuits and there is a hard tally language in NP?co-NP, then there is a Turing complete language for NP that is not many-one complete.  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.
np-completeness length-increasing reductions turing completeness computational complexity approximable sets