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Optimal constant in n^{c·d+o(d)} for L1 and L∞ EMDuT

Determine the optimal constant c>0 such that computing Earth Mover’s Distance under Translation (EMDuT) for the L1 and L∞ metrics in dimension d≥2 can be solved in time n^{c·d+o(d)}.

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Background

For L1 and L∞ in higher dimensions, the paper presents algorithms with running time \widetilde{O}(m{d}n{d+2}) and ETH‑based lower bounds ruling out n{o(d)}‑time (1+ε)‑approximations, indicating a tight exponential‑in‑d dependence up to constants.

The open issue is to pin down the precise constant in the exponent governing the best achievable running time as a function of d.

References

For the L_1 and L_\infty metric in dimension d \ge 2 we leave open to determine the optimal constant c>0 such that the problem can be solved in time n{c\cdot d + o(d)}.

— 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 L1 and L∞ metric in higher dimensions.”