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DOI: 10.4230/LIPIcs.STACS.2011.649
URN: urn:nbn:de:0030-drops-30514
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2011/3051/
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Knauer, Christian ; Tiwary, Hans Raj ; Werner, Daniel

On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems

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Abstract

We study several canonical decision problems arising from some well-known theorems from combinatorial geometry. Among others, we show that computing the minimum size of a Caratheodory set and a Helly set and certain decision versions of the hs cut problem are W[1]-hard (and NP-hard) if the dimension is part of the input. This is done by fpt-reductions (which are actually ptime-reductions) from the d-Sum problem. Our reductions also imply that the problems we consider cannot be solved in time n^{o(d)} (where n is the size of the input), unless the Exponential-Time Hypothesis (ETH) is false.

The technique of embedding d-Sum into a geometric setting is conceptually much simpler than direct fpt-reductions from purely combinatorial W[1]-hard problems (like the clique problem) and has great potential to show (parameterized) hardness and (conditional) lower bounds for many other problems.

BibTeX - Entry

@InProceedings{knauer_et_al:LIPIcs:2011:3051,
  author =	{Christian Knauer and Hans Raj Tiwary and Daniel Werner},
  title =	{{On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
  pages =	{649--660},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Thomas Schwentick and Christoph D{\"u}rr},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2011/3051},
  URN =		{urn:nbn:de:0030-drops-30514},
  doi =		{10.4230/LIPIcs.STACS.2011.649},
  annote =	{Keywords: computational geometry, combinatorial geometry, ham-sandwich cuts, parameterized complexity.}
}

Keywords: computational geometry, combinatorial geometry, ham-sandwich cuts, parameterized complexity.
Collection: 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)
Issue Date: 2011
Date of publication: 11.03.2011

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