When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2010.2462
URN: urn:nbn:de:0030-drops-24627
URL: http://drops.dagstuhl.de/opus/volltexte/2010/2462/
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### Collapsing and Separating Completeness Notions under Average-Case and Worst-Case Hypotheses

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### Abstract

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.

### BibTeX - Entry

@InProceedings{gu_et_al:LIPIcs:2010:2462,
author =	{Xiaoyang Gu and John M. Hitchcock and Aduri Pavan},
title =	{{Collapsing and Separating Completeness Notions under Average-Case and Worst-Case Hypotheses}},
booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
pages =	{429--440},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-16-3},
ISSN =	{1868-8969},
year =	{2010},
volume =	{5},
editor =	{Jean-Yves Marion and Thomas Schwentick},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},