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arXiv · 1205.6314

Minimal external representations of tropical polyhedra

Xavier Allamigeon·Ricardo D. Katz
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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".

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Xavier Allamigeon, Ricardo D. Katz. 2013-02-13. Minimal external representations of tropical polyhedra. https://doi.org/10.1016/j.jcta.2013.01.011

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