The Fr\'echet Contingency Array Problem is Max-Plus Linear

TL;DR

This paper demonstrates that the array Fréchet problem in probability and statistics is max-plus linear, providing new insights into its structure and solution methods through residuation theory and greedy algorithms.

Contribution

It reveals the max-plus linearity of the Fréchet problem and introduces a greedy algorithm leveraging this property and the Monge condition.

Findings

Upper bound derived via residuation theory

Lower bound established as a greedy loop invariant

Algorithm exploits max-plus linearity and Monge property

Abstract

In this paper we show that the so-called array Fr\'echet problem in Probability/Statistics is (max, +)-linear. The upper bound of Fr\'echet is obtained using simple arguments from residuation theory and lattice distributivity. The lower bound is obtained as a loop invariant of a greedy algorithm. The algorithm is based on the max-plus linearity of the Fr\'echet problem and the Monge property of bivariate distribution.