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

Abstract

Let $x_1<\dots0, $$ 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.