Cops and Robbers from a distance

Bonato, Anthony; Chiniforooshan, Ehsan; Praat, Pawe

doi:10.1016/j.tcs.2010.07.003

Cited by 35 publications

(32 citation statements)

References 11 publications

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“…There exist many open problems in these two games and in this paper, we define and study a generalization of these games. We believe that our game can lead to results of some of these open problems, much like other variants (see [6,12,2,8,10]) which have been studied for the same purpose. For example, a strategy for the guards in our game may be adapted to a strategy for the guards in the eternal domination game.…”

Section: Introductionmentioning

confidence: 79%

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Spy-game on graphs: Complexity and simple topologies

Cohen¹,

Martins²,

Inerney³

et al. 2018

Theoretical Computer Science

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We define and study the following two-player game on a graph G. Let k ∈ 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 ∈ N * is his speed. Then, each guard may move along one edge. The spy and the guards may occupy the same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d ∈ 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 (for every speed s and distance d) and that some variant of it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards, the speed s of the spy and the required distance d when G is a path or a cycle.

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Section: Introductionmentioning

confidence: 79%

“…The Cops and robber game has been generalized in many ways [6,12,2,8,10]. In [6], Bonato et al proposed a variant with radius of capture. That is, the cops win if one of them reaches a vertex at distance at most d (a fixed integer) from the robber.…”

Section: Related Workmentioning

confidence: 99%

Spy-game on graphs: Complexity and simple topologies

Cohen¹,

Martins²,

Inerney³

et al. 2018

Theoretical Computer Science

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“…Several variants have been studied such as when the cops and the robber have different speeds [FGK + 10, CCNV11, AM10, Meh11], when the robber can be captured at some distance [BCP10], when each cop can be moved a bounded number of time [FGL10], etc. In the variant proposed in [FGH + 08,FGL09], the goal for the cops is to guard some part of a graph, i.e., to prevent the robber to reach some particular vertices in the graph.…”

Section: Cops and Robber Gamesmentioning

confidence: 99%

To satisfy impatient Web surfers is hard

Fomin¹,

Giroire²,

Jean-Marie³

et al. 2014

Theoretical Computer Science

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Prefetching is a basic mechanism to avoid to waste time when accessing data. However, a tradeoff must be established 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 user is surfing on the Web. 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 and that ensures that all documents accessed by the web surfer have previously been loaded in the cache.We model this problem as a game similar to Cops and Robber Games in graphs. A fugitive starts on a marked vertex of a (di)graph G. Turn by turn, an observer marks at most k ≥ 1 vertices and then the fugitive can move along one edge/arcs of G. The observer wins if he prevents the fugitive to reach an unmarked vertex. The fugitive wins otherwise, i.e., if she enters an unmarked vertex. The surveillance number of a graph is the least k ≥ 1 allowing the observer to win whatever the fugitive does. 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.All our results hold for both variants, connected or not. We show that deciding whether the surveillance number of a chordal graph equals 2 is NP-hard. Deciding if the surveillance number of a DAG equals 4 is PSPACEcomplete. Moreover, computing the surveillance number is NP-hard in split graphs. On the other hand, we provide polynomial time algorithms to compute surveillance number of trees and interval graphs. Moreover, in the case of trees, we establish a combinatorial characterization, related to isoperimetry, of the surveillance number. Un fugitif débute sur un sommet initialement marqué d'un graphe (orienté) G. Alors, tour-à-tour un surveillant marque au plus k sommets de G et le fugitif peut se déplacer le long d'une arête (d'un arc) de G. Le surveillant gagne si iĺ evite toujours que le fugitif atteigne un sommet non marqué. Le fugitif gagne dans le cas contraire. L'indice de contrôle d'un graphe (orienté) G est le plus petit entier k ≥ 1 qui permet au surveillant de gagner quels que soient les déplacements du fugitif. Nous considéronségalement la variante connexe de ce jeu, dans laquelle, le surveillant ne peut marquer que des voisins de sommets préalablement marqués.Tous nos résultats sont valides pour les deux variantes (connexe ou non) du jeu. Nous prouvons que décider si l'indice de contrôle d'un graphe cordal vaut 2 est NP-difficile. En particulier, le problème de décision associéà l 'indice de contrôle n'est pas FPT. Puis, nous montrons que calculer l'indice de contrôle est NP-difficile dans la classe des split graphes (une sous-classe des graphes cordaux). Dans le cas des graphes orientés, nous montrons que décider si l'indice de contrôle d'un DAG vaut 4 est PSPACE-complet.Nous présentons ensuite un algorithme exponentiel exact qui...

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“…Indeed, there is a rich literature on these games under various names [13], such as man-and-the-lion [12,16,9], cops-and-robber [6,1,8,3,4], robot-and-rabbit [6], and pursuit-evasion [14,5,7], just to name a few.…”

Section: Introductionmentioning

confidence: 99%

“…This makes sense in situations where the defenders (cops) have fixed patrol routes, but not in interactive games like kabaddi. The problems and results in the graph searching literature are also of a different nature than ours [2,11], although variations using differential speed [4] and capture from a distance [3] have been considered in graphs as well.…”

Section: Introductionmentioning

confidence: 99%

Multiagent Pursuit Evasion, or Playing Kabaddi

Klein

Suri

2010

Springer Tracts in Advanced Robotics

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We study a version of pursuit evasion where two or more pursuers are required to capture the evader because the evader is able to overpower a single defender. The pursuers must coordinate their moves to fortify their approach against the evader while the evader maneuvers to disable pursuers from their unprotected sides. We model this situation as a game of Kabaddi, a popular South Asian sport where two teams occupy opposite halves of a field and take turns sending an attacker into the other half, in order to win points by tagging or wrestling members of the opposing team, while holding his breath during the attack. The game involves team coordination and movement strategies, making it non-trivial to formally model and analyze, yet provides an elegant framework for the study of multiagent pursuitevasion, for instance, a team of robots attempting to capture a rogue agent. Our paper introduces a simple discrete (time and space) model for the game, offers analysis of winning strategies, and explores tradeoffs between maximum movement speed, number of pursuers, and locational constraints. 1

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Cops and Robbers from a distance

Cited by 35 publications

References 11 publications

Spy-game on graphs: Complexity and simple topologies

Spy-game on graphs: Complexity and simple topologies

To satisfy impatient Web surfers is hard

Multiagent Pursuit Evasion, or Playing Kabaddi

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