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

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 $\max_{σ\in S_n} F_σ(x,y) := \frac12 \sum_{i,k=1}^n |x_i - x_k|^α\, |y_{σ(i)} - y_{σ(k)}|^α, \quad α>0,$ is in general neither solved by the identical permutation (id) nor the anti-identical permutation (a-id) if $n > 2 +2^α$. 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.

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