Characterization of tropical hemispaces by(P,R)-decompositions

Abstract

We consider tropical hemispaces, defined as tropically convex sets whose complements are also tropically convex, and tropical semispaces, defined as maximal tropically convex sets not containing a given point. We introduce the concept of $(P,R)$-decomposition. This yields (to our knowledge) a new kind of representation of tropically convex sets extending the classical idea of representing convex sets by means of extreme points and rays. We characterize tropical hemispaces as tropically convex sets that admit a (P,R)-decomposition of certain kind. In this characterization, with each tropical hemispace we associate a matrix with coefficients in the completed tropical semifield, satisfying an extended rank-one condition. Our proof techniques are based on homogenization (lifting a convex set to a cone), and the relation between tropical hemispaces and semispaces.Comment: 29 pages, 3 figure

Minimizing maximum lateness in two-stage projects by tropical optimization

“…We now evaluate the computational complexity of the solution of the upperlevel problem. First, we consider the condition at (18). The most computationally demanding part of the condition is the calculation of the function Tr(Q) and the Kleene star matrices Q * and S * .…”

“…Since the calculation of the Kleene star matrix using Floyd-Warshall algorithm has the polynomial computational complexity of the third degree, the condition at (18) requires O(min(m, n) 3 ) operations.…”

The geometric structure of max-plus hemispaces

“…b) The k-faces of minimal codimension appeared before in the literature under the name of sectors. They are related to the complements of semispaces [4,7,11,12,13,8].…”

Tropical Fourier–Motzkin elimination, with an application to real-time verification

“…tropically convex sets whose complements are also tropically convex, which were studied in e.g. [BH08,KNS13]), see Theorem 13 and Corollary 17.…”

“…In contrast, non-closed tropically convex sets may not be finitely generated. Generating representations of (non-necessarily closed) tropical convex cones have been studied in [BSS07], and in [KNS13] in the particular case of tropical hemispaces. A certain class of possibly infinite generating representations was treated in [GK04], however, the associated algorithms rely on the expensive Presburger arithmetic.…”