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Study of a Combinatorial Game in Graphs Through Linear Programming

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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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Notes

  1. For any graph G, any integer \(\ell \) and \(v\in V(G)\), let \(N_{\ell }[v]\) be the set of vertices at distance at most \(\ell \) from v in G and let \(N[v]=N_1[v]\).

  2. Indeed, \(O((n/d)^2)\) vertices are sufficient to dominate every vertex at distance d in \(G_{n \times n}\) (tiling the grid with vertex-disjoint balls of radius d).

  3. In strategy \(P_{2r_1}\), that guard must be at distance \(\le d\) from \((2r_1,2j_1)\) when the spy visits it.

  4. Similarly, for strategy \(P_{2r_2}\).

  5. Solving the LP for \(n\ge 150\) takes more than one hour on a basic laptop.

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Authors and Affiliations

  1. LRI, CNRS, Univ Paris Sud, Orsay, France

    Nathann Cohen

  2. I3S, Université Côte d’Azur, Inria, CNRS, Sophia Antipolis, France

    Fionn Mc Inerney, Nicolas Nisse & Stéphane Pérennes

Authors
  1. Nathann Cohen
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  2. Fionn Mc Inerney
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  3. Nicolas Nisse
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  4. Stéphane Pérennes
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Correspondence to Nicolas Nisse.

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This work has been partially supported by ANR project Stint under reference ANR-13-BS02-0007, ANR program “Investments for the Future” under reference ANR-11- LABX-0031-01, the associated Inria team AlDyNet. Extended abstracts of parts of this paper have been presented in [8] (Sect. 5.1) and [9].

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Cohen, N., Mc Inerney, F., Nisse, N. et al. Study of a Combinatorial Game in Graphs Through Linear Programming. Algorithmica 82, 212–244 (2020). https://doi.org/10.1007/s00453-018-0503-9

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