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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.

Cite as

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-dev.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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