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DOI: 10.4230/LIPIcs.STACS.2011.649
URN: urn:nbn:de:0030-drops-30514
URL: https://drops.dagstuhl.de/opus/volltexte/2011/3051/
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. | |
| Seminar: | 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) | |
| Issue date: | 2011 | |
| Date of publication: | 11.03.2011 |
