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Spy-Game on Graphs

Authors Nathann Cohen, Mathieu Hilaire, Nícolas A. Martins, Nicolas Nisse, Stéphane Pérennes



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Nathann Cohen
Mathieu Hilaire
Nícolas A. Martins
Nicolas Nisse
Stéphane Pérennes

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Nathann Cohen, Mathieu Hilaire, Nícolas A. Martins, Nicolas Nisse, and Stéphane Pérennes. Spy-Game on Graphs. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 10:1-10:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.FUN.2016.10

Abstract

We define and study the following two-player game on a graph G. Let k in N^*. A set of k guards is occupying some vertices of G while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where s in N^* is his speed. Then, each guard may move along one edge. The spy and the guards may occupy same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d in N (a predefined distance) from every guard. Can the spy win against k guards? Similarly, what is the minimum distance d such that k guards may ensure that at least one of them remains at distance at most d from the spy? This game generalizes two well-studied games: Cops and robber games (when s=1) and Eternal Dominating Set (when s is unbounded). We consider the computational complexity of the problem, showing that it is NP-hard and that it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards and the required distance d when G is a path or a cycle. Our main result is that there exists beta>0 such that Omega(n^{1+beta}) guards are required to win in any n*n grid.
Keywords
  • graph
  • two-player games
  • cops and robber games
  • complexity

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References

  1. M. Aigner and M. Fromme. A game of cops and robbers. Discrete Applied Mathematics, 8:1-12, 1984. Google Scholar
  2. N. Alon and A. Mehrabian. On a generalization of Meyniel’s conjecture on the cops and robbers game. Electr. J. Comb., 18(1), 2011. Google Scholar
  3. A. Bonato, E. Chiniforooshan, and P. Pralat. Cops and robbers from a distance. Theor. Comput. Sci., 411(43):3834-3844, 2010. Google Scholar
  4. A. Bonato and R. Nowakovski. The game of Cops and Robber on Graphs. American Math. Soc., 2011. Google Scholar
  5. J. Chalopin, V. Chepoi, N. Nisse, and Y. Vaxès. Cop and robber games when the robber can hide and ride. SIAM J. Discrete Math., 25(1):333-359, 2011. Google Scholar
  6. Uriel Feige. A threshold of log n for approximating set cover. J. ACM, 45(4):634-652, 1998. Google Scholar
  7. F. V. Fomin, P. A. Golovach, J. Kratochvíl, N. Nisse, and K. Suchan. Pursuing a fast robber on a graph. Theor. Comput. Sci., 411(7-9):1167-1181, 2010. Google Scholar
  8. F. V. Fomin, P. A. Golovach, and D. Lokshtanov. Cops and robber game without recharging. In 12th Scandinavian Symp. and Workshops on Algorithm Theory (SWAT), volume 6139 of LNCS, pages 273-284. Springer, 2010. Google Scholar
  9. F.V. Fomin, P. A. Golovach, and P. Pralat. Cops and robber with constraints. SIAM J. Discrete Math., 26(2):571-590, 2012. Google Scholar
  10. W. Goddard, S.M. Hedetniemi, and S.T. Hedetniemi. Eternal security in graphs. J. Combin.Math.Combin.Comput., 52, 2005. Google Scholar
  11. John L. Goldwasser and William Klostermeyer. Tight bounds for eternal dominating sets in graphs. Discrete Mathematics, 308(12):2589-2593, 2008. Google Scholar
  12. William B. Kinnersley. Cops and robbers is exptime-complete. J. Comb. Theory, Ser. B, 111:201-220, 2015. Google Scholar
  13. W.F. Klostermeyer and G MacGillivray. Eternal dominating sets in graphs. J. Combin.Math.Combin.Comput., 68, 2009. Google Scholar
  14. W.F. Klostermeyer and C.M. Mynhardt. Graphs with equal eternal vertex cover and eternal domination numbers. Discrete Mathematics, 311(14):1371-1379, 2011. Google Scholar
  15. R. J. Nowakowski and P. Winkler. Vertex-to-vertex pursuit in a graph. Discrete Maths, 43:235-239, 1983. Google Scholar
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