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Study of a Combinatorial Game in Graphs Through Linear Programming

Authors
  • Cohen, Nathann
  • Mc Inerney, Fionn
  • Nisse, Nicolas
  • Pérennes, Stéphane
Publication Date
Aug 31, 2018
Source
HAL-UPMC
Keywords
Language
English
License
Unknown
External links

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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