Convex Hulls, Oracles, and Homology

This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the ``no''-case of POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).Comment: 11 pages, 2 figure

Vertex-Facet Incidences of Unbounded Polyhedra

How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a polyhedron that is simple and "simplicial," i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices on each facet, we derive from the structure of the vertex-facet incidence matrix that the polyhedron is necessarily bounded. In particular, this yields a characterization of those polyhedra that have circulants as vertex-facet incidence matrices.Comment: LaTeX2e, 14 pages with 4 figure

Construction and Analysis of Projected Deformed Products

We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are ``strictly preserved'' under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose projection to the last dcoordinates yields a neighborly cubical d-polytope. As an extension of thecubical case, we construct matrix representations of deformed products of(even) polygons (DPPs), which have a projection to d-space that retains the complete (\lfloor \tfrac{d}{2} \rfloor - 1)-skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the ``projected deformed products of polygons'' that were announced by Ziegler (2004), a family of 4-polytopes whose ``fatness'' gets arbitrarily close to 9.Comment: 20 pages, 5 figure

Prodsimplicial-Neighborly Polytopes

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k.Comment: 28 pages, 9 figures; minor correction

Mechanics and force transmission in soft composites of rods in elastic gels

We report detailed theoretical investigations of the micro-mechanics and bulk elastic properties of composites consisting of randomly distributed stiff fibers embedded in an elastic matrix in two and three dimensions. Recent experiments published in Physical Review Letters [102, 188303 (2009)] have suggested that the inclusion of stiff microtubules in a softer, nearly incompressible biopolymer matrix can lead to emergent compressibility. This can be understood in terms of the enhancement of the compressibility of the composite relative to its shear compliance as a result of the addition of stiff rod-like inclusions. We show that the Poisson's ratio $\nu$ of such a composite evolves with increasing rod density towards a particular value, or {\em fixed point}, independent of the material properties of the matrix, so long as it has a finite initial compressibility. This fixed point is $\nu=1/4$ in three dimensions and $\nu=1/3$ in two dimensions. Our results suggest an important role for stiff filaments such as microtubules and stress fibers in cell mechanics. At the same time, our work has a wider elasticity context, with potential applications to composite elastic media with a wide separation of scales in stiffness of its constituents such as carbon nanotube-polymer composites, which have been shown to have highly tunable mechanics.Comment: 10 pages, 8 figure

Minimal half-spaces and external representation of tropical polyhedra

We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.Comment: 19 pages, 4 figures, example added with a new figure, figures improved, references update

Combinatorial simplex algorithms can solve mean payoff games

A combinatorial simplex algorithm is an instance of the simplex method in which the pivoting depends on combinatorial data only. We show that any algorithm of this kind admits a tropical analogue which can be used to solve mean payoff games. Moreover, any combinatorial simplex algorithm with a strongly polynomial complexity (the existence of such an algorithm is open) would provide in this way a strongly polynomial algorithm solving mean payoff games. Mean payoff games are known to be in NP and co-NP; whether they can be solved in polynomial time is an open problem. Our algorithm relies on a tropical implementation of the simplex method over a real closed field of Hahn series. One of the key ingredients is a new scheme for symbolic perturbation which allows us to lift an arbitrary mean payoff game instance into a non-degenerate linear program over Hahn series.Comment: v1: 15 pages, 3 figures; v2: improved presentation, introduction expanded, 18 pages, 3 figure

Tropical polyhedra are equivalent to mean payoff games

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory.Comment: 28 pages, 5 figures; v2: updated references, added background materials and illustrations; v3: minor improvements, references update