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Connected Surveillance Game
"... Abstract: The surveillance game [Fomin et al., 2012] models the problem of webpage prefetching as a pursuit evasion game played on a graph. This twoplayer game is played turnbyturn. The first player, called the observer, can mark a fixed amount of vertices at each turn. The second one controls a ..."
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Abstract: The surveillance game [Fomin et al., 2012] models the problem of webpage prefetching as a pursuit evasion game played on a graph. This twoplayer game is played turnbyturn. The first player, called the observer, can mark a fixed amount of vertices at each turn. The second one controls a surfer that stands at vertices of the graph and can slide along edges. The surfer starts at some initially marked vertex of the graph, her objective is to reach an unmarked node before all nodes of the graph are marked. The surveillance number sn(G) of a graph G is the minimum amount of nodes that the observer has to mark at each turn ensuring it wins against any surfer in G. Fomin et al. also defined the connected surveillance game where the observer must ensure that marked nodes always induce a connected subgraph. They ask what is the cost of connectivity, i.e., is there a constant c> 0 such that the ratio between the connected surveillance number csn(G) and sn(G) is at most c for any graph G. It is straightforward to show that csn(G) ≤ ∆ sn(G) for any graph G with maximum degree ∆. Moreover, it has been shown that there are graphs G for which csn(G) = sn(G)+1. In this paper, we investigate the question of the cost of the connectivity. We first provide new nontrivial upper and lower bounds for the cost of connectivity in the surveillance game. More precisely, we present a family of graphs G such that csn(G)> sn(G) + 1.
Fractional Combinatorial Games on Graphs
 15ÈMES RENCONTRES FRANCOPHONES SUR LES ASPECTS ALGORITHMIQUES DES TÉLÉCOMMUNICATIONS (ALGOTEL) (2013)
, 2013
"... De nombreux jeux impliquant deux joueurs dans un graphe ont été étudiés en théorie des graphes: Gendarmes et voleur, Ange et Démon, Observeur et surfeur, Dominants universels, etc. Outre la capture d’un fugitif ou la lutte contre le feu, ces jeux ont aussi des applications dans les réseaux de téléco ..."
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De nombreux jeux impliquant deux joueurs dans un graphe ont été étudiés en théorie des graphes: Gendarmes et voleur, Ange et Démon, Observeur et surfeur, Dominants universels, etc. Outre la capture d’un fugitif ou la lutte contre le feu, ces jeux ont aussi des applications dans les réseaux de télécommunications car, d’une part, ils permettent de mieux appréhender les structures des réseaux, et d’autre part, ils permettent de modéliser et d’étudier des problèmes de ces réseaux (e.g., problème de cache dans l’internet). Dans tous ces jeux, chaque joueur contrôle des jetons sur les sommets du graphe et selon les jeux, les joueurs peuvent: déplacer des jetons le long des arêtes du graphe, ajouter/supprimer des jetons, etc. Dans ce travail, nous proposons une approche générale en définissant un jeu qui constitue, entre autre, une relaxation fractionnaire de tous les jeux mentionnés cidessus. Pour ce jeu générique, nous montrons qu’il existe un algorithme en temps polynomial, en le nombre de sommets du graphe et le nombre de maximum de tours de jeu autorisés, pour décider si un des joueurs a une stratégie gagnante. Cet algorithme permet de calculer une stratégie efficace (approximation à un facteur logn près), gagnante avec forte probabilité, pour le problème de cache.
THEME
"... 5.1.1. Epidemic models of propagation of content 3 5.1.2. Control and game models for malware attack 3 5.1.3. Time random walks on time varying graphs 3 5.1.4. Quick detection of central nodes 3 5.1.5. Graphbased semisupervised learning 3 5.1.6. Optimal weight selection in average consensus protoc ..."
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5.1.1. Epidemic models of propagation of content 3 5.1.2. Control and game models for malware attack 3 5.1.3. Time random walks on time varying graphs 3 5.1.4. Quick detection of central nodes 3 5.1.5. Graphbased semisupervised learning 3 5.1.6. Optimal weight selection in average consensus protocols 4 5.1.7. Reducing communication overhead of average consensus protocols 4 5.2. Wireless Networks 4 5.2.1. Estimation of population sizes in sensor networks 4
Fractional Combinatorial Twoplayer Games
, 2013
"... During the last decades, many combinatorial games involving two persons playing on a (directed) graph have received a lot of attention. Some examples of such games are the Angel problem, the Cops and Robbers, the Surveillance game, the Eternal Dominating Set and Eternal Set Cover. One of the main ..."
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During the last decades, many combinatorial games involving two persons playing on a (directed) graph have received a lot of attention. Some examples of such games are the Angel problem, the Cops and Robbers, the Surveillance game, the Eternal Dominating Set and Eternal Set Cover. One of the main questions in these games is to decide if a given player has a winning strategy. That is, if it can always win regardless of behaviour of the other player. This question is often NPhard. In the Cops and Robbers game and for the Surveillance game this question is PSPACEcomplete [Mamino 2012, Fomin et al. 2012]. In this paper, we propose a fractional relaxation of these games. That is, we present a framework, based on linear programming techniques, that can be used to model any of the aforementioned games and some of its variants. As far as we know, it is the first time that such combinatorial games have been studied in this way and perspectives are promising. We also propose an algorithm that decides whether the first player can win the game in at most t turns. Moreover, under a weak assumption which is valid for all the aforementioned games, the fractional game gives us a lower bound for the integral game. For the Surveillange game and the Angel problem we show that there is, with high probability, a winning strategy which is in a O(log n)