Fast approximation algorithms for 1D EMDuT

Determine whether there exist constant‑factor or (1+ε)‑approximation algorithms for Earth Mover’s Distance under Translation (EMDuT) on the real line that run in strongly subquadratic time O(n^{2−δ}) for some constant δ>0 or in near‑linear time, and specifically whether a (1+ε)‑approximation can be computed in time \widetilde{O}(n^{2−δ}/poly(ε)) for some constant δ>0 (independent of n and ε) or even in time \widetilde{O}(n/poly(ε)).

Background

The paper gives near‑quadratic algorithms and OVH‑based lower bounds for asymmetric EMDuT in one dimension, ruling out (1+ε)‑approximations in O(n^{2−δ}/ε^{o(1)}) time for any constant δ>0 when |B|=Ω(n). However, the existence of faster approximation algorithms in 1D remains unresolved.

This question targets whether one can beat the quadratic barrier with either constant‑factor approximations or (1+ε)‑approximations, potentially even achieving near‑linear time with controlled ε‑dependence.