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

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.