On Assignment Problems Related to Gromov-Wasserstein Distances on the Real Line

TL;DR

This paper investigates assignment problems related to Gromov-Wasserstein distances on the real line, revealing that common solutions like identity or anti-identity permutations are not always optimal for certain parameters and point configurations.

Contribution

It provides a counterexample showing that standard permutations do not always solve the assignment problem connected to Gromov-Wasserstein distances, highlighting the complexity of these problems.

Findings

Standard permutations may not solve the assignment problem for certain parameters.

The maximum value of the assignment function can be arbitrarily far from the identity or anti-identity solutions.

The study emphasizes the complexity of Gromov-Wasserstein related assignment problems.

Abstract

Let $x_1 < \dots < x_n$ and $y_1 < \dots < y_n$, $n \in \mathbb N$, be real numbers. We show by an example that the assignment problem $$ \max_{\sigma \in S_n} F_\sigma(x,y) := \frac12 \sum_{i,k=1}^n |x_i - x_k|^\alpha \, |y_{\sigma(i)} - y_{\sigma(k)}|^\alpha, \quad \alpha >0, $$ is in general neither solved by the identical permutation (id) nor the anti-identical permutation (a-id) if $n > 2 +2^\alpha$. Indeed the above maximum can be, depending on the number of points, arbitrary far away from $F_\text{id}(x,y)$ and $F_\text{a-id}(x,y)$. The motivation to deal with such assignment problems came from their relation to Gromov-Wasserstein divergences which have recently attained a lot of attention.