Study of a combinatorial game in graphs through Linear Programming
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.
Domains
Discrete Mathematics [cs.DM]
Origin : Files produced by the author(s)
Loading...