Sub-three approximation in truly subquadratic time

Determine whether a (3−ε)-approximation for ordinary edit distance can be obtained in truly subquadratic time for some absolute constant ε>0, and more ambitiously whether a (1+ε)-approximation can be obtained in truly subquadratic time for every fixed ε>0.

Background

The paper establishes a factor-three approximation for ordinary edit distance in truly subquadratic time, improving prior guarantees that were slightly above three. The authors identify the precise threshold below factor three as unresolved and note that their center-correction technique only compensates for losses that vanish with the input length.

The authors also point out that the approximation scheme of Mao and Rubinstein provides a different step toward a (1+ε)-approximation, but its running time is not truly subquadratic. No known fine-grained lower bound rules out such an approximation even in nearly linear time.

References

Can one obtain a $(3-\varepsilon)$-approximation in truly subquadratic time for some absolute constant $\varepsilon>0$? Our correction pays for losses that vanish with the input length. Saving a fixed fraction of the total cost would require new ideas. The approximation scheme of Mao and Rubinstein makes a step in this direction from a different starting point but their time complexity of $N2/2{\log{\Omega(1)}N}$ is not truly subquadratic. More ambitiously, can one obtain a $(1+\varepsilon)$-approximation in truly subquadratic time for every fixed $\varepsilon>0$? No known fine-grained lower bound even excludes $(1+\varepsilon)$-approximation in $(N)$ time for any fixed positive $\varepsilon$.

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

Can the sampling in these algorithms be replaced so as to obtain a deterministic constant-factor approximation in truly subquadratic time? This remains a natural question even for unit costs.

— Factor Three Approximation for Edit Distance  (2610.01311 - Gorbachev, 1 Oct 2026) in Section 1, subsection “Open Questions,” item 4

Can the $(1+\varepsilon)$-approximation of Mao--Rubinstein be extended to arbitrary metric weights with a saving of the form $2{\log{\Omega(1)}N}$ over quadratic time? Such a result would address a different tradeoff from the truly subquadratic factor-three algorithms considered here. Logarithmic savings are known for exact weighted edit distance over a constant-size alphabet: Crochemore, Landau, and Ziv-Ukelson achieve $(N2/\log N)$ time with unrestricted real-valued scoring matrices. This does not give an alphabet-independent speedup in our metric-oracle model. A weaker target is a $(1+\varepsilon)$-approximation in $(N2/\logc N)$ time for some constant $c>0$ and every fixed $\varepsilon>0$, independently of the alphabet size and the numerical range of the costs.

— Factor Three Approximation for Edit Distance  (2610.01311 - Gorbachev, 1 Oct 2026) in Section 1, subsection “Open Questions,” item 5