Fine-Grained Complexity of Earth Mover's Distance under Translation

The Earth Mover’s Distance is a popular similarity measure in several branches of computer science. It measures the minimum total edge length of a perfect matching between two point sets. The Earth Mover’s Distance under Translation ($\mathrm{E}\mathrm{M}\mathrm{D}\mathrm{u}\mathrm{T}$) is a translation-invariant version thereof. It minimizes the Earth Mover’s Distance over all translations of one point set. For $\mathrm{E}\mathrm{M}\mathrm{D}\mathrm{u}\mathrm{T}$ in ${\mathbb{R}}^1$, we present an $\tilde{\mathcal{O}}(n^2)$-time algorithm. We also show that this algorithm is nearly optimal by presenting a matching conditional lower bound based on the Orthogonal Vectors Hypothesis. For $\mathrm{E}\mathrm{M}\mathrm{D}\mathrm{u}\mathrm{T}$ in ${\mathbb{R}}^d$, we present an $\tilde{\mathcal{O}}(n^{2d+2})$-time algorithm for the $L_1$ and $L_{\mathrm{\infty }}$ metric. We show that this dependence on $d$ is asymptotically tight, as an $n^{o(d)}$-time algorithm for $L_1$ or $L_{\mathrm{\infty }}$ would contradict the Exponential Time Hypothesis (ETH). Prior to our work, only approximation algorithms were known for these problems.