ER‑hardness of Euclidean Traveling Salesperson Problem (ETSP)

Determine whether the Euclidean Traveling Salesperson Problem—given n planar points with rational coordinates and a rational threshold t, decide whether there exists a tour of total Euclidean length less than t—is hard for the class ER (exists‑R) under polynomial‑time reductions.

Background

The authors show how ETSP fits into ER but explicitly note that its ER‑hardness status is unknown. Establishing ER‑hardness would align ETSP with other geometric problems complete for ER, while a non‑hardness result could indicate a finer‑grained structure within ER.

This question probes whether ETSP captures the full expressive difficulty of existential real arithmetic or occupies a potentially easier subclass.