Band-width spanner dilation trade-off

Establish whether every finite point set in \mathbb{R}^d admits a geometric spanner of band-width $k$ and dilation $O(n/k^{d/(d-1)})$.

Background

The paper establishes asymptotically tight dilation bounds of O(n/kd/(d−1))O(n/k^{d/(d-1)}) for spanners bounded by several width parameters, including tree-width, path-width, branch-width, cut-width, directed tree-width, directed path-width, and DAG-width. Band-width is closely related to path-width but is more restrictive than cut-width, and the paper does not provide a corresponding upper bound for band-width-bounded spanners. The unresolved issue is whether the same asymptotically optimal trade-off extends to this parameter for every point set in Rd\mathbb{R}^d.

References

One parameter that remains open is band-width: This is closely related to path-width, but even more restrictive than cut-width. Thus it is open, whether there exists a spanner of band-width $k$ and dilation $O(n/k{d/(d-1)})$ for every set of points in $\mathbb{R}d$.

— On (Directed) Width-Parameters of Geometric Spanners  (2609.22082 - Buchin et al., 18 Sep 2026) in Section Conclusion and Open Questions; Table in Appendix, Section Tabular Overview of Dilation Bounds for Different Graph Parameters