Tropical polar cones, hypergraph transversals, and mean payoff games

Details Tropical polar cones, hypergraph transversals, and mean payoff games Tropical polar cones, hypergraph transversals, and mean payoff games

2010-04-16

arxiv.org/abs/1004.2778v2

Linear Algebra and its Applications, Volume 435, Issue 7, 1 October 2011, Pages 1549-1574

Mathematics

Combinatorics

27 pages, 6 figures, revised version

Scientific paper

10.1016/j.laa.2011.02.004

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.

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