Tropical polar cones, hypergraph transversals, and mean payoff games

TL;DR

This paper explores tropical convexity, characterizing polar cones, extreme rays, and implications for linear inequalities using hypergraph transversals and mean payoff games, revealing structural properties of tropical polyhedral cones.

Contribution

It introduces tropical analogues of convex duality concepts, characterizes extreme rays via minimal set covers, and links inequality implication to mean payoff game strategies.

Findings

Number of extreme rays of tropical cyclic polyhedral cone is polynomially bounded.

No unique minimal system of inequalities defines a tropical polyhedral cone.

Tropical Farkas lemma provides a game-theoretic certificate for inequality implication.

Abstract

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.