269 research outputs found

    On the computational complexity of a game of cops and robbers

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    We study the computational complexity of a perfect-information two-player game proposed by Aigner and Fromme (1984) [1]. The game takes place on an undirected graph where n simultaneously moving cops attempt to capture a single robber, all moving at the same speed. The players are allowed to pick their starting positions at the first move. The question of the computational complexity of deciding this game was raised by Goldstein and Reingold (1995) [9]. We prove that the game is hard for PSPACE.©2012 Elsevier B.V. All rights reserved

    Visibility Graphs, Dismantlability, and the Cops and Robbers Game

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    We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit-evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs.Comment: 23 page

    Cops and Robbers is EXPTIME-complete

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    We investigate the computational complexity of deciding whether k cops can capture a robber on a graph G. In 1995, Goldstein and Reingold conjectured that the problem is EXPTIME-complete when both G and k are part of the input; we prove this conjecture.Comment: v2: updated figures and slightly clarified some minor point

    Characterizations and algorithms for generalized Cops and Robbers games

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    We propose a definition of generalized Cops and Robbers games where there are two players, the Pursuer and the Evader, who each move via prescribed rules. If the Pursuer can ensure that the game enters into a fixed set of final positions, then the Pursuer wins; otherwise, the Evader wins. A relational characterization of the games where the Pursuer wins is provided. A precise formula is given for the length of the game, along with an algorithm for computing if the Pursuer has a winning strategy whose complexity is a function of the parameters of the game. For games where the position of one player does not affect the available moves of he other, a vertex elimination ordering characterization, analogous to a cop-win ordering, is given for when the Pursuer has a winning strategy

    Dagstuhl Reports : Volume 1, Issue 2, February 2011

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    Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-HĂŒbner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro PezzĂ©, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

    Localization game on geometric and planar graphs

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    The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph GG we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant ζ(G)\zeta (G), defined as the least number of cops needed to localize the robber on a graph GG, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth 22 and unbounded ζ(G)\zeta (G). On a positive side, we prove that ζ(G)\zeta (G) is bounded by the pathwidth of GG. We then show that the algorithmic problem of determining ζ(G)\zeta (G) is NP-hard in graphs with diameter at most 22. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane
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