1,101 research outputs found
Cop and robber games when the robber can hide and ride
International audienceIn the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a ïŹnite graph G = (V , E). The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is called cop win if the cop always captures the robber after a ïŹnite number of steps. Nowakowski, Winkler (1983) and Quilliot (1983) characterized the cop-win graphs as dismantlable graphs. In this talk, we will characterize in a similar way the class CWFR(s, sâČ ) of cop-win graphs in the game in which the cop and the robber move at different speeds sâČ and s, sâČ â€ s. We also establish some connections between cop-win graphs for this game with sâČ 1. We characterize the graphs which are k-winnable for any value of k
Cop and robber game and hyperbolicity
In this note, we prove that all cop-win graphs G in the game in which the
robber and the cop move at different speeds s and s' with s'<s, are
\delta-hyperbolic with \delta=O(s^2). We also show that the dependency between
\delta and s is linear if s-s'=\Omega(s) and G obeys a slightly stronger
condition. This solves an open question from the paper (J. Chalopin et al., Cop
and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25
(2011) 333-359). Since any \delta-hyperbolic graph is cop-win for s=2r and
s'=r+2\delta for any r>0, this establishes a new - game-theoretical -
characterization of Gromov hyperbolicity. We also show that for weakly modular
graphs the dependency between \delta and s is linear for any s'<s. Using these
results, we describe a simple constant-factor approximation of the
hyperbolicity \delta of a graph on n vertices in O(n^2) time when the graph is
given by its distance-matrix
Characterizations and algorithms for generalized Cops and Robbers games
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
Cops and Invisible Robbers: the Cost of Drunkenness
We examine a version of the Cops and Robber (CR) game in which the robber is
invisible, i.e., the cops do not know his location until they capture him.
Apparently this game (CiR) has received little attention in the CR literature.
We examine two variants: in the first the robber is adversarial (he actively
tries to avoid capture); in the second he is drunk (he performs a random walk).
Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD),
which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being
the expected capture times in the adversarial and drunk CiR variants,
respectively. We show that these capture times are well defined, using game
theory for the adversarial case and partially observable Markov decision
processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD
for several special graph families such as -regular trees, give some bounds
for grids, and provide general upper and lower bounds for general classes of
graphs. We also give an infinite family of graphs showing that iCOD can be
arbitrarily close to any value in [2,infinty). Finally, we briefly examine one
more CiR variant, in which the robber is invisible and "infinitely fast"; we
argue that this variant is significantly different from the Graph Search game,
despite several similarities between the two games
Dagstuhl Reports : Volume 1, Issue 2, February 2011
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
Spy-Game on Graphs
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
Spy-Game on graphs
International audienceWe define and study the following two-player game on a graph G. Let k â 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 â 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 â 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 > 0 such that âŠ(n 1+) guards are required to win in any n Ă n grid
k-Chordal Graphs: from Cops and Robber to Compact Routing via Treewidth
International audienceCops and robber games, introduced by Winkler and Nowakowski [41] and independently defined by Quilliot [43], concern a team of cops that must capture a robber moving in a graph. We consider the class of k-chordal graphs, i.e., graphs with no induced (chordless) cycle of length greater than k, k â„ 3. We prove that k â 1 cops are always sufficient to capture a robber in k-chordal graphs. This leads us to our main result, a new structural decomposition for a graph class including k-chordal graphs. We present a polynomial-time algorithm that, given a graph G and k â„ 3, either returns an induced cycle larger than k in G, or computes a tree-decomposition of G, each bag of which contains a dominating path with at most k â 1 vertices. This allows us to prove that any k-chordal graph with maximum degree â has treewidth at most (k â1)(â â1) +2, improving the O(â (â â1) kâ3) bound of Bodlaender and Thilikos (1997). Moreover, any graph admitting such a tree-decomposition has small hyperbolicity. As an application, for any n-vertex graph admitting such a tree-decomposition, we propose a compact routing scheme using routing tables, addresses and headers of size O(k log â + log n) bits and achieving an additive stretch of O(k log â). As far as we know, this is the first routing scheme with O(k log â + log n)-routing tables and small additive stretch for k-chordal graphs
To satisfy impatient Web surfers is hard
International audiencePrefetching is a basic mechanism for faster data access and efficient computing. An important issue in prefetching is the tradeoff between the amount of network's resources wasted by the prefetching and the gain of time. For instance, in the Web, browsers may download documents in advance while a Web surfer is surfing. Since the Web surfer follows the hyperlinks in an unpredictable way, the choice of the Web pages to be prefetched must be computed online. The question is then to determine the minimum amount of resources used by prefetching that ensures that all documents accessed by theWeb surfer have previously been loaded in the cache. We model this problem as a two-player game similar to Cops and Robber Games in graphs. Let k 1 be any integer. The first player, a fugitive, starts on a marked vertex of a (di)graph G. The second player, an observer, marks at most k vertices, then the fugitive moves along one edge/arc of G to a new vertex, then the observer marks at most k vertices, etc. The fugitive wins if it enters an unmarked vertex, and the observer wins otherwise. The surveillance number of a (di)graph is the minimum k such that the observer marking at most k vertices at each step can win against any strategy of the fugitive. We also consider the connected variant of this game, i.e., when a vertex can be marked only if it is adjacent to an already marked vertex. We study the computational complexity of the game. All our results hold for both variants, connected or unrestricted. We show that deciding whether the surveillance number of a chordal graph is at most 2 is NP-hard. We also prove that deciding if the surveillance number of a DAG is at most 4 is PSPACEcomplete. Moreover, we show that the problem of computing the surveillance number is NP-hard in split graphs. On the other hand, we provide polynomial time algorithms computing surveillance numbers of trees and interval graphs. Moreover, in the case of trees, we establish a combinatorial characterization of the surveillance number
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