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Study of a combinatorial game in graphs through Linear Programming

Nathann Cohen 1 Fionn Mc Inerney 2 Nicolas Nisse 2 Stéphane Pérennes 2
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 torus 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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Contributor : Nicolas Nisse <>
Submitted on : Tuesday, September 5, 2017 - 3:37:04 PM
Last modification on : Monday, December 14, 2020 - 3:44:02 PM


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  • HAL Id : hal-01462890, version 2


Nathann Cohen, Fionn Mc Inerney, Nicolas Nisse, Stéphane Pérennes. Study of a combinatorial game in graphs through Linear Programming. [Research Report] Inria Sophia Antipolis. 2017. ⟨hal-01462890v2⟩



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