Related papers: Long and winding central paths
We show that finding the classical bound of broad families of Bell inequalities can be naturally framed as the contraction of an associated tensor network, but in tropical algebra, where the sum is replaced by the minimum and the product is…
A bounded curvature path is a continuously differentiable piece-wise $C^2$ path with bounded absolute curvature connecting two points in the tangent bundle of a surface. These paths have been widely considered in computer science and…
We introduce tropical complexes, as an enrichment of the dual complex of a degeneration with additional data from non-transverse intersection numbers. We define cycles, divisors, and linear equivalence on tropical complexes, analogous both…
The main result of this paper is a formula for the limit cycle of a 1-parameter family of subvarieties of a tropical compactification, expressed in terms of tropical intersections. Our theorem generalizes results of…
In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the…
We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated…
We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.
We present simple compact proofs of the strong and weak duality theorems of tropical linear programming. It follows that there is no duality gap for a pair of tropical primal-dual problems. This result together with known properties of…
We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral…
For many common height functions, it is notoriously hard to compute the essential minimum. Nevertheless there are two classical methods, one giving lower bounds and the other giving upper bounds. In this paper, we show that the two methods…
In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…
In this work, we relate girth and path-degeneracy in classes with sub-exponential expansion, with explicit bounds for classes with polynomial expansion and proper minor-closed classes that are tight up to a constant factor (and tight up to…
We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the…
This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of…
We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument. It is possible to show, interpreting the following notions appropriately, that if…
In this text, we merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism…
It is proved that, for an indefinite quadratic programming problem under linear constraints, any iterative sequence generated by the Proximal DC decomposition algorithm $R$-linearly converges to a Karush-Kuhn-Tucker point, provided that the…
The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still open whether the Hirsch…
In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert's Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric…
We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows…