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

TL;DR

This work initiates a fine-grained complexity study of the Earth Mover's Distance under Translation (EMDuT). It delivers a near-optimal 1D algorithm with time $\tilde{O}(n^2)$ for EMDuT in $\mathbb{R}^1$ and a matching OVH-based lower bound, while extending to higher dimensions with $\tilde{O}(m^d n^{d+2}\log^{d+2} n)$ algorithms for $L_1$ and $L_\infty$. It further establishes ETH-based lower bounds ruling out $n^{o(d)}$-time approximations in higher dimensions and presents conditional reductions from $k$-Clique to show inherent hardness in the symmetric cases. By combining a sweep-line approach, a refined data-structure (Overmars–van Leeuwen) extension, and arrangement-based techniques, the paper clarifies the complexity landscape of EMDuT, separates exact/approximation barriers, and lays a foundation for future work on the $L_2$ case. Collectively, these results bridge exact algorithm design and strong conditional hardness for translation-invariant geometric transport problems.

Abstract

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 ($\mathrm{EMDuT}$) is a translation-invariant version thereof. It minimizes the Earth Mover's Distance over all translations of one point set. For $\mathrm{EMDuT}$ 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 $\mathrm{EMDuT}$ in $\mathbb{R}^d$, we present an $\tilde{\mathcal{O}}(n^{2d+2})$-time algorithm for the $L_1$ and $L_\infty$ metric. We show that this dependence on $d$ is asymptotically tight, as an $n^{o(d)}$-time algorithm for $L_1$ or $L_\infty$ would contradict the Exponential Time Hypothesis (ETH). Prior to our work, only approximation algorithms were known for these problems.

Figures (11)

• Figure 1.1: Given a set of (solid) blue points $B$ and a set of red points $R$, our goal is to find a translation $\tau$ (shown in green) and a perfect matching from $B+\tau$ to $R$ (shown in black) that minimizes the total distance of matched pairs.
• Figure 3.1: Schematic representation of the graph $G = \phi \oplus \phi'$ used in the proof of \ref{['lem:move_forward']}. Each edge exists if and only if exactly one edge from either $\phi$ or $\phi'$ is present. Green edges arise from the matching $\phi'$, while yellow edges arise from the matching $\phi$.
• Figure 3.2: Case when connected component of $G$ is a cycle.
• Figure 3.3: Two cases of \ref{['claim:increasing-seq']} in which a crossing occurs.
• Figure 3.4: Illustration of crossing types. The left figure shows a BBRR crossing and the right figure shows a RBBR crossing.

Theorems & Definitions (55)

• Theorem 1.1: 1D Algorithms
• Theorem 1.2: 1D Lower Bound
• Theorem 1.3: Algorithms for $L_1$ and $L_\infty$ metric, Asymmetric
• Theorem 1.4: Lower Bound for $L_1$ and $L_\infty$ metric, Symmetric
• Lemma 3.0
• proof
• Theorem 3.2
• Lemma 3.2
• proof
• Claim 3.3