Study of a Combinatorial Game in Graphs Through Linear Programming

Nathann Cohen 1 Fionn Mc Inerney 2 Nicolas Nisse 2 Stéphane Pérennes 2
1 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
LRI - Laboratoire de Recherche en Informatique
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In the Spy game played on a graph G, a single spy travels the vertices of G at speed s, while multiple slow guards strive to have, at all times, one of them within distance d of that spy. In order to determine the smallest number of guards necessary for this task, we analyze the game through a Linear Programming formulation and the fractional strategies it yields for the guards. We then show the equivalence of fractional and integral strategies in trees. This allows us to design a polynomial-time algorithm for computing an optimal strategy in this class of graphs. Using duality in Linear Programming, we also provide non-trivial bounds on the fractional guard-number of grids and tori, which gives a lower bound for the integral guard-number in these graphs. We believe that the approach using fractional relaxation and Linear Programming is promising to obtain new results in the field of combinatorial games.
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https://hal.archives-ouvertes.fr/hal-01881473
Contributeur : Nathann Cohen <>
Soumis le : mercredi 10 octobre 2018 - 16:22:40
Dernière modification le : lundi 5 novembre 2018 - 15:36:03

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spygame_trees_grids_July18.pdf
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Nathann Cohen, Fionn Mc Inerney, Nicolas Nisse, Stéphane Pérennes. Study of a Combinatorial Game in Graphs Through Linear Programming. Algorithmica, Springer Verlag, In press, 〈10.1007/s00453-018-0503-9〉. 〈hal-01881473〉

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