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.ISAAC.2017.22
URN: urn:nbn:de:0030-drops-82254
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8225/
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Cohen, Nathann ; Mc Inerney, Fionn ; Nisse, Nicolas ; Pérennes, Stéphane

Study of a Combinatorial Game in Graphs Through Linear Programming

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LIPIcs-ISAAC-2017-22.pdf (0.7 MB)

Abstract

In the Spy Game played on a graph G, a single spy travels the ertices of G at speed s, while multiple slow guards strive to have, at all times, one of them within distance d of that spy. In order to determine the smallest number of guards necessary for this task, we analyze the game through a Linear Programming formulation and the fractional strategies it yields for the guards. We then show the equivalence of fractional and integral strategies in trees. This allows us to design a polynomial-time algorithm for computing an optimal strategy in this class of graphs. Using duality in Linear Programming, we also provide non-trivial bounds on the fractional guardnumber of grids and torus. We believe that the approach using fractional relaxation and Linear Programming is promising to obtain new results in the field of combinatorial games.

BibTeX - Entry

@InProceedings{cohen_et_al:LIPIcs:2017:8225,
  author =	{Nathann Cohen and Fionn Mc Inerney and Nicolas Nisse and St{\'e}phane P{\'e}rennes},
  title =	{{Study of a Combinatorial Game in Graphs Through Linear Programming}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{22:1--22:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Yoshio Okamoto and Takeshi Tokuyama},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8225},
  URN =		{urn:nbn:de:0030-drops-82254},
  doi =		{10.4230/LIPIcs.ISAAC.2017.22},
  annote =	{Keywords: Turn-by-turn games in graphs, Graph algorithms, Linear Programming}
}

Keywords: Turn-by-turn games in graphs, Graph algorithms, Linear Programming
Collection: 28th International Symposium on Algorithms and Computation (ISAAC 2017)
Issue Date: 2017
Date of publication: 07.12.2017

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