NP‑membership 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 in NP on a standard Turing/word‑RAM model.

Background

The paper asks readers to place ETSP within ER (exists‑R) and then explicitly notes that its NP‑membership is unknown. The subtlety arises from verifying sums of square roots in polynomial time, a long‑standing issue tied to the exact arithmetic of Euclidean distances.

Clarifying NP‑membership would connect geometric optimization more tightly with discrete verification frameworks and resolve a long‑standing complexity classification gap for ETSP.

References

It is unknown whether the $Euclidean Travelling Salesperson Problem$ is in .

— Beyond Bits: An Introduction to Computation over the Reals  (2603.29427 - Miltzow, 31 Mar 2026) in Section “Existential Theory of the Reals,” Exercises, item on Euclidean Travelling Salesperson Problem, subitem (b)