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DOI: 10.4230/LIPIcs.STACS.2011.649

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

Authors: Christian Knauer, Hans Raj Tiwary, and Daniel Werner

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

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.

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Christian Knauer, Hans Raj Tiwary, and Daniel Werner. On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 649-660, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{knauer_et_al:LIPIcs.STACS.2011.649,
  author =	{Knauer, Christian and Tiwary, Hans Raj and Werner, Daniel},
  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 =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.649},
  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.}
}

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

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