Quasi-polynomial running-time savings for Fréchet distance

Establish whether computing or approximating the Fréchet distance between polygonal curves admits a quasi-polynomial improvement in running time over the standard quadratic-time algorithms, despite known SETH-based hardness precluding polynomial savings.

Background

The Fréchet distance is a classical similarity measure between curves. Bringmann (FOCS 2014) proves SETH-based conditional lower bounds ruling out strongly subquadratic algorithms (and certain approximation improvements), which imply no polynomial running-time savings are possible under these assumptions.

In contrast to the string problems addressed in this paper, where quasi-strongly subquadratic improvements are achieved via randomized approximation schemes, it remains unresolved whether similar quasi-polynomial savings in running time are possible for the Fréchet distance.

References

For the Frechet distance, existing hardness of approximation results rule out polynomial savings assuming SETH , and it remains open whether quasi-polynomial saving is possible.

— Approximation Schemes for Edit Distance and LCS in Quasi-Strongly Subquadratic Time  (2603.29702 - Mao et al., 31 Mar 2026) in Additional related Work (un-numbered subsection under Introduction)