Faster factor-three edit-distance algorithms

Determine whether the exponents 11/6 for unit-cost edit distance and 40/21 for arbitrary metric-weighted edit distance can be reduced, ideally toward the N^{8/5+o(1)} running times known for approximation factors just above three, and establish whether arbitrary metric weights require any additional polynomial overhead over unit costs.

Background

The paper gives an N{11/6}-time factor-three algorithm for unit-cost edit distance and an N{40/21}-time factor-three algorithm for arbitrary metric edit weights. These bounds improve the approximation factor to exactly three but are slower than existing algorithms whose factors are just above three.

The authors specifically ask whether the metric-weighted setting inherently incurs a polynomial overhead compared with the unit-cost setting.

References

Can the exponents $11/6$ and $40/21$ be reduced? A natural target is to approach the $N{8/5+o(1)}$ running times available for factors just above three, for both unit and arbitrary metric costs. It is also unclear whether arbitrary metrics must incur any additional polynomial overhead over the unit-cost case.

— Factor Three Approximation for Edit Distance  (2610.01311 - Gorbachev, 1 Oct 2026) in Section 1, subsection “Open Questions,” item 2

Does there exist an absolute constant $C$ such that $C$-approximation of edit distance can be achieved in $N{1+o(1)}$ time? Andoni--Nosatzki allow every fixed exponent $1+\delta$, but their approximation factor depends on $\delta$. For arbitrary metric weights, even an analogue of their fixed-$\delta$ tradeoff would be valuable.

— Factor Three Approximation for Edit Distance  (2610.01311 - Gorbachev, 1 Oct 2026) in Section 1, subsection “Open Questions,” item 3