Distance-Query Version of the Estimation Method

Develop a distance-query version of the Held–Karp and graph-metric TSP estimation method for metrics accessed through distance queries rather than insertion-only weighted graph streams.

Background

The results apply to connected positive-weight graphs presented as insertion-only edge streams. The paper notes that the approach does not directly yield a sublinear-query algorithm for metrics specified through distance queries, and explicitly leaves such a version open.

References

Tour construction and a distance-query version remain open for this method.

— Semi-Streaming Estimation of the Held--Karp Bound and Graph-Metric TSP Cost  (2610.18705 - Alipour, 13 Oct 2026) in Section 5, “Limitations and open questions” (Section \ref{sec:limits})