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Closing the gap for (1+ε)-approximation of L2 EMDuT

Close the gap between the quadratic lower bound and the exponential upper bound for (1+ε)-approximation algorithms for Earth Mover’s Distance under Translation (EMDuT) under the L2 metric, specifically for algorithms whose running time has a 1/ε^{o(1)} dependency on ε.

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Background

For L2, the paper highlights a large gap: the only known lower bound (from 1D) rules out O(n{2−δ}/ε{o(1)}) time, while a naive approach via fixing matchings yields an exponential‑time (1+ε) approximation n{O(n)} log3(1/ε) for constant d.

The authors explicitly call for resolving this discrepancy either by faster algorithms or stronger lower bounds under the stated ε‑dependence.

References

We pose as an open problem to close this huge gap between the quadratic lower and exponential upper bound (for (1+\varepsilon)-approximation algorithms with a 1/\varepsilon{o(1)} dependency on \varepsilon in the running time).

— 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.”