Approximation Schemes for Edit Distance and LCS in Quasi-Strongly Subquadratic Time
arXiv:2603.29702v1 Announce Type: new Abstract: We present novel randomized approximation schemes for the Edit Distance (ED) problem and the Longest Common Subsequence (LCS) problem that, for any constant $\epsilon>0$, compute a $(1+\epsilon)$-approximation for ED and a $(1-\epsilon)$-approximation for LCS in time $n^2 / 2^{\log^{\Omega(1)}(n)}$ 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,
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Abstract:We present novel randomized approximation schemes for the Edit Distance (ED) problem and the Longest Common Subsequence (LCS) problem that, for any constant $\epsilon>0$, compute a $(1+\epsilon)$-approximation for ED and a $(1-\epsilon)$-approximation for LCS in time $n^2 / 2^{\log^{\Omega(1)}(n)}$ 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.
Comments: Accepted to STOC 2026
Subjects:
Data Structures and Algorithms (cs.DS)
Cite as: arXiv:2603.29702 [cs.DS]
(or arXiv:2603.29702v1 [cs.DS] for this version)
https://doi.org/10.48550/arXiv.2603.29702
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Xiao Mao [view email] [v1] Tue, 31 Mar 2026 12:57:29 UTC (269 KB)
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