170 research outputs found
Tropical Convexity
The notions of convexity and convex polytopes are introduced in the setting
of tropical geometry. Combinatorial types of tropical polytopes are shown to be
in bijection with regular triangulations of products of two simplices.
Applications to phylogenetic trees are discussed.
Theorem 29 and Corollary 30 in the paper, relating tropical polytopes to
injective hulls, are incorrect. See the erratum at
http://www.math.uiuc.edu/documenta/vol-09/vol-09-eng.html .Comment: 20 pages, 6 figure
Splitting Polytopes
A split of a polytope is a (regular) subdivision with exactly two maximal
cells. It turns out that each weight function on the vertices of admits a
unique decomposition as a linear combination of weight functions corresponding
to the splits of (with a split prime remainder). This generalizes a result
of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite
metric spaces.
Introducing the concept of compatibility of splits gives rise to a finite
simplicial complex associated with any polytope , the split complex of .
Complete descriptions of the split complexes of all hypersimplices are
obtained. Moreover, it is shown that these complexes arise as subcomplexes of
the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].Comment: 25 pages, 7 figures; minor corrections and change
A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine
Combining insights from the study of type refinement systems and of monoidal
closed chiralities, we show how to reconstruct Lawvere's hyperdoctrine of
presheaves using a full and faithful embedding into a monoidal closed
bifibration living now over the compact closed category of small categories and
distributors. Besides revealing dualities which are not immediately apparent in
the traditional presentation of the presheaf hyperdoctrine, this reconstruction
leads us to an axiomatic treatment of directed equality predicates (modelled by
hom presheaves), realizing a vision initially set out by Lawvere (1970). It
also leads to a simple calculus of string diagrams (representing presheaves)
that is highly reminiscent of C. S. Peirce's existential graphs for predicate
logic, refining an earlier interpretation of existential graphs in terms of
Boolean hyperdoctrines by Brady and Trimble. Finally, we illustrate how this
work extends to a bifibrational setting a number of fundamental ideas of linear
logic.Comment: Identical to the final version of the paper as appears in proceedings
of LICS 2016, formatted for on-screen readin
On duality and fractionality of multicommodity flows in directed networks
In this paper we address a topological approach to multiflow (multicommodity
flow) problems in directed networks. Given a terminal weight , we define a
metrized polyhedral complex, called the directed tight span , and
prove that the dual of -weighted maximum multiflow problem reduces to a
facility location problem on . Also, in case where the network is
Eulerian, it further reduces to a facility location problem on the tropical
polytope spanned by . By utilizing this duality, we establish the
classifications of terminal weights admitting combinatorial min-max relation
(i) for every network and (ii) for every Eulerian network. Our result includes
Lomonosov-Frank theorem for directed free multiflows and
Ibaraki-Karzanov-Nagamochi's directed multiflow locking theorem as special
cases.Comment: 27 pages. Fixed minor mistakes and typos. To appear in Discrete
Optimizatio
- …