Related papers: Tropical determinant on transportation polytope
This paper presents a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it…
In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependence allows for a natural definition of matrix rank in a sense that coincides with the notions of…
In this paper, we study the complexity of the periodic temporal graph realization problem with respect to upper bounds on the fastest path durations among its vertices. This constraint with respect to upper bounds appears naturally in…
Mathematical modeling is a standard approach to solve many real-world problems and {\em diversity} of solutions is an important issue, emerging in applying solutions obtained from mathematical models to real-world problems. Many studies…
We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method is based on a numerical resolution of the corresponding…
In this paper, we introduce the Maximum Matrix Contraction problem, where we aim to contract as much as possible a binary matrix in order to maximize its density. We study the complexity and the polynomial approximability of the problem.…
In this article, we consider the monomial complete intersection algebra $\mathbb{K}[x,y]/\langle x^d,y^q\rangle$ in two variables. For elements $l_1,\ldots,l_{d+q-2k}$ of degree $1$, we give a formula of the deteminant of linear map from…
Let $\{w_{i,j}\}_{1\leq i\leq n, 1\leq j\leq s} \subset L_m=F(X_1,...,X_m)[{\partial \over \partial X_1},..., {\partial \over \partial X_m}]$ be linear partial differential operators of orders with respect to ${\partial \over \partial…
Let $T(n)$ denote the maximum number of unit distances that a set of $n$ points in the Euclidean plane $\mathbb{R}^2$ can determine with the additional condition that the distinct unit length directions determined by the configuration must…
We describe an approach for finding upper bounds on an ODE dynamical system's maximal Lyapunov exponent among all trajectories in a specified set. A minimization problem is formulated whose infimum is equal to the maximal Lyapunov exponent,…
The nonnegative and positive semidefinite (PSD-) ranks are closely connected to the nonnegative and positive semidefinite extension complexities of a polytope, which are the minimal dimensions of linear and SDP programs which represent this…
We derive a linear programming bound on the maximum cardinality of error-correcting codes in the sum-rank metric. Based on computational experiments on relatively small instances, we observe that the obtained bounds outperform all…
This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its…
We study tropical degree bounds, stable tropical intersections, and tropical B\'ezout-type estimates through the geometry of Newton polytopes, mixed subdivisions, and lattice indices. We establish an upper bound for the tropical degree of a…
Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer…
In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
Among $\R^3$-valued triples of random vectors $(X,Y,Z)$ having fixed marginal probability laws, what is the best way to jointly draw $(X,Y,Z)$ in such a way that the simplex generated by $(X,Y,Z)$ has maximal average volume? Motivated by…
The space T_{d,n} of n tropically collinear points in a fixed tropical projective space TP^{d-1} is equivalent to the tropicalization of the determinantal variety of matrices of rank at most 2, which consists of real d x n matrices of…
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…