pp 1–33 | Cite as

Study of a Combinatorial Game in Graphs Through Linear Programming

  • Nathann Cohen
  • Fionn Mc Inerney
  • Nicolas Nisse
  • Stéphane Pérennes
Part of the following topical collections:
  1. Special Issue: Algorithms and Computation


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.


Cops and robber games Spy game Domination Graphs Linear Programming Tree Grid 



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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LRICNRS, Univ Paris SudOrsayFrance
  2. 2.I3SUniversité Côte d’Azur, Inria, CNRSSophia AntipolisFrance

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