Using stochastic order to compare different Euclidean Random Assignment Problems

TL;DR

This paper introduces a theorem leveraging stochastic order to compare minimum total costs in Euclidean Random Assignment Problems, with applications to one-dimensional and higher-dimensional cases, enhancing understanding of cost relationships.

Contribution

The paper presents a novel theorem that uses stochastic order to compare costs in Euclidean Random Assignment Problems, expanding analytical tools in this domain.

Findings

Theorem for comparing costs using stochastic order.

Applications to 1D $k$-star graph problems.

Comparisons of problems in higher dimensions.

Abstract

This paper provides a theorem to compare the minimum total cost of two different Euclidean Random Assignment Problems with the same number of points, using the stochastic order of the costs of one of the pairs in these two problems. The subsequent sections provide two applications of the theorem, including studies of the problem on the one-dimensional $k$-star graph and comparisons between some problems in higher dimensions. More possible applications and limitations of the theorem are also discussed.