Minimal external representations of tropical polyhedra

TL;DR

This paper proves that tropical polyhedral cones have essentially unique minimal external representations, characterized by a partial anti-exchange property, with apices linked to the associated cell complex vertices.

Contribution

It introduces the concept of minimal external representations for tropical polyhedra and establishes their uniqueness and structural properties.

Findings

Existence of essentially unique minimal external representations.

Characterization of apices as vertices of the associated cell complex.

A necessary and sufficient condition for vertices to be apices of non-redundant half-spaces.

Abstract

Tropical polyhedra are known to be representable externally, as intersections of finitely many tropical half-spaces. However, unlike in the classical case, the extreme rays of their polar cones provide external representations containing in general superfluous half-spaces. In this paper, we prove that any tropical polyhedral cone in R^n (also known as "tropical polytope" in the literature) admits an essentially unique minimal external representation. The result is obtained by establishing a (partial) anti-exchange property of half-spaces. Moreover, we show that the apices of the half-spaces appearing in such non-redundant external representations are vertices of the cell complex associated with the polyhedral cone. We also establish a necessary condition for a vertex of this cell complex to be the apex of a non-redundant half-space. It is shown that this condition is sufficient for a dense class of polyhedral cones having "generic extremities".