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Complexity of EMDuT under the L2 metric in higher dimensions

Determine the computational complexity of Earth Mover’s Distance under Translation (EMDuT) under the Euclidean L2 metric in any dimension d≥2.

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Background

Unlike L1/L∞, the paper notes that exact computation for L2 EMDuT in d≥2 is impossible due to the geometric median’s algebraic hardness, and current techniques neither yield n{O(d)} algorithms nor rule out n{o(d)} time.

The state of the art only provides a 1D lower bound excluding O(n{2−δ}/ε{o(1)})‑time (1+ε) approximations and a trivial exponential‑in‑n upper bound via enumerating matchings, leaving the overall complexity unresolved.

References

The L_2 metric is the most natural measure in the geometric settings, making _2 a well motivated problem. The most pressing open problem is to determine the complexity of the _2 problem in any dimension d \ge 2.

— Fine-Grained Complexity of Earth Mover's Distance under Translation (2403.04356 - Bringmann et al., 7 Mar 2024) in Section Open problems, subparagraph “EMDuT for L2 metric in higher dimensions.”