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

Abstract

Let and , , be real numbers. We show by an example that the assignment problem \begin{align*} \max_{\sigma \in S_n} F_\sigma (x,y) := \frac 12 \sum_{i,k=1}^n |x_i- x_k|^\alpha \, |y_{\sigma (i)}- y_{\sigma (k)}|^\alpha, \quad \alpha \gt 0, \end{align*} is in general neither solved by the identical permutation nor the anti-identical permutation if . Indeed the above maximum can be, depending on the number of points, arbitrarily far away from and . The motivation to deal with such assignment problems came from their relation to Gromov–Wasserstein distances, which have recently received a lot of attention in imaging and shape analysis.