License: Creative Commons Attribution-NoDerivs 3.0 Unported license (CC BY-ND 3.0)
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://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2462/
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Gu, Xiaoyang ; Hitchcock, John M. ; Pavan, Aduri

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},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2462},
  URN =		{urn:nbn:de:0030-drops-24627},
  doi =		{10.4230/LIPIcs.STACS.2010.2462},
  annote =	{Keywords: Computational complexity, NP-completeness}
}

Keywords: Computational complexity, NP-completeness
Collection: 27th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2010
Date of publication: 09.03.2010

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