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DOI: 10.4230/LIPIcs.CCC.2017.6
URN: urn:nbn:de:0030-drops-75151
URL: http://drops.dagstuhl.de/opus/volltexte/2017/7515/
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Nguyen, Danny ; Pak, Igor

The Computational Complexity of Integer Programming with Alternations

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LIPIcs-CCC-2017-6.pdf (0.6 MB)


Abstract

We prove that integer programming with three alternating quantifiers is NP-complete, even for a fixed number of variables. This complements earlier results by Lenstra and Kannan, which together say that integer programming with at most two alternating quantifiers can be done in polynomial time for a fixed number of variables. As a byproduct of the proof, we show that for two polytopes P, Q in R^4, counting the projection of integer points in Q\P is #P-complete. This contrasts the 2003 result by Barvinok and Woods, which allows counting in polynomial time the projection of integer points in P and Q separately.

BibTeX - Entry

@InProceedings{nguyen_et_al:LIPIcs:2017:7515,
  author =	{Danny Nguyen and Igor Pak},
  title =	{{The Computational Complexity of Integer Programming with Alternations}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{Ryan O'Donnell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7515},
  URN =		{urn:nbn:de:0030-drops-75151},
  doi =		{10.4230/LIPIcs.CCC.2017.6},
  annote =	{Keywords: Integer Programming, Alternations, Projection of Integer Points}
}

Keywords: Integer Programming, Alternations, Projection of Integer Points
Seminar: 32nd Computational Complexity Conference (CCC 2017)
Issue Date: 2017
Date of publication: 21.07.2017


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