Collapsing and Separating Complete Notions Under Worst-Case and Average-Case Hypotheses
Xiaofeng Gu, John M. Hitchcock, A. Pavan · 2010
This paper presents the following results on sets that are complete for NP. (i) If there is a problem in NP that requires 2nΩ(1) time at almost all lengths, then every manyone 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 averagecase and worst-case hypotheses in the last result is unique as previous results obtaining the same consequence relied on almost-everywhere hardness results. 1