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 (EMDuT) is a translation-invariant version thereof. It minimizes the Earth Mover’s Distance over all translations of one point set. For EMDuT in R¹, we present an O^e(n²)-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 EMDuT in R^d, we present an O^e(n²d⁺²)-time algorithm for the L1 and L∞ metric. We show that this dependence on d is asymptotically tight, as an n^o(d)-time algorithm for L1 or L∞ would contradict the Exponential Time Hypothesis (ETH). Prior to our work, only approximation algorithms were known for these problems.