License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2018.9
URN: urn:nbn:de:0030-drops-94725
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9472/
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Brandt, Sebastian ; Pettie, Seth ; Uitto, Jara

Fine-grained Lower Bounds on Cops and Robbers

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LIPIcs-ESA-2018-9.pdf (0.6 MB)

Abstract

Cops and Robbers is a classic pursuit-evasion game played between a group of g cops and one robber on an undirected N-vertex graph G. We prove that the complexity of deciding the winner in the game under optimal play requires Omega (N^{g-o(1)}) time on instances with O(N log^2 N) edges, conditioned on the Strong Exponential Time Hypothesis. Moreover, the problem of calculating the minimum number of cops needed to win the game is 2^{Omega (sqrt{N})}, conditioned on the weaker Exponential Time Hypothesis. Our conditional lower bound comes very close to a conditional upper bound: if Meyniel's conjecture holds then the cop number can be decided in 2^{O(sqrt{N}log N)} time.
In recent years, the Strong Exponential Time Hypothesis has been used to obtain many lower bounds on classic combinatorial problems, such as graph diameter, LCS, EDIT-DISTANCE, and REGEXP matching. To our knowledge, these are the first conditional (S)ETH-hard lower bounds on a strategic game.

BibTeX - Entry

@InProceedings{brandt_et_al:LIPIcs:2018:9472,
  author =	{Sebastian Brandt and Seth Pettie and Jara Uitto},
  title =	{{Fine-grained Lower Bounds on Cops and Robbers}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{9:1--9:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Yossi Azar and Hannah Bast and Grzegorz Herman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9472},
  URN =		{urn:nbn:de:0030-drops-94725},
  doi =		{10.4230/LIPIcs.ESA.2018.9},
  annote =	{Keywords: Cops and Robbers}
}

Keywords: Cops and Robbers
Collection: 26th Annual European Symposium on Algorithms (ESA 2018)
Issue Date: 2018
Date of publication: 14.08.2018

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