Approximation Schemes for Edit Distance and LCS in Quasi-Strongly Subquadratic Time

2026-03-31

AI Summary

This work investigates efficient approximation algorithms for edit distance (ED) and longest common subsequence (LCS), breaking the quadratic-time barrier inherent in classical dynamic programming. The authors present the first randomized algorithm that, for any constant $\varepsilon > 0$, achieves a $(1+\varepsilon)$-approximation for ED and a $(1-\varepsilon)$-approximation for LCS in strongly subquadratic time $n^2 / 2^{\log^{\Omega(1)} n}$. This result establishes a new benchmark for the trade-off between time complexity and approximation accuracy, while also revealing a fundamental distinction between approximate and exact computation in the context of fine-grained complexity. Furthermore, the paper demonstrates an intrinsic derandomization barrier for approximating LCS, suggesting that randomness may be essential for achieving such subquadratic approximations.

Abstract

We present novel randomized approximation schemes for the Edit Distance (ED) problem and the Longest Common Subsequence (LCS) problem that, for any constant $ε>0$, compute a $(1+ε)$-approximation for ED and a $(1-ε)$-approximation for LCS in time $n^2 / 2^{\log^{Ω(1)}(n)}$ for two strings of total length at most $n$. This running time improves upon the classical quadratic-time dynamic programming algorithms by a quasi-polynomial factor. Our results yield significant insights into fine-grained complexity: Firstly, for ED, prior work indicates that any exact algorithm cannot be improved beyond a few logarithmic factors without refuting established complexity assumptions [Abboud, Hansen, Vassilevska Williams, Williams, 2016]; our quasi-polynomial speed-up shows a separation the complexity of approximate ED from that of exact ED, even for approximation factor arbitrarily close to $1$. Secondly, for LCS, obtaining similar approximation-time tradeoffs via deterministic algorithms would imply breakthrough circuit lower bounds [Chen, Goldwasser, Lyu, Rothblum, Rubinstein, 2019]; our randomized algorithm demonstrates derandomization hardness for LCS approximation.

Problem

• Edit Distance
• Longest Common Subsequence
• Approximation Algorithms
• Fine-Grained Complexity
• Subquadratic Time

Innovation

• Edit Distance
• Longest Common Subsequence
• Randomized Approximation
• Fine-Grained Complexity
• Quasi-Polynomial Speedup