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

Abstract : 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 all lengths, then every many-one NP-complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. 2. 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. 3. If there exist a one-way permutation that is secure against subexponential-size circuits and there is a hard tally language in NP intersect coNP, 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.
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Jean-Yves Marion and Thomas Schwentick. 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Mar 2010, Nancy, France. pp.429-440, 2010, Proceedings of the 27th Annual Symposium on the Theoretical Aspects of Computer Science
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Xiaoyang Gu, John M. Hitchcock, Aduri Pavan. Collapsing and Separating Completeness Notions under Average-Case and Worst-Case Hypotheses. Jean-Yves Marion and Thomas Schwentick. 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Mar 2010, Nancy, France. pp.429-440, 2010, Proceedings of the 27th Annual Symposium on the Theoretical Aspects of Computer Science. <inria-00455731>

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