Approximation Algorithms for LCS and LIS with Truly Improved Running Times

TL;DR

This paper introduces novel approximation algorithms for the Longest Common Subsequence (LCS) and Longest Increasing Subsequence (LIS) problems that operate in truly subquadratic and sublinear time respectively, with approximation factors depending on the solution-to-input size ratio.

Contribution

The work presents the first algorithms achieving truly subquadratic and sublinear time for approximating LCS and LIS without prior knowledge of the solution ratio, extending triangle inequality techniques to non-metric settings.

Findings

Truly subquadratic time algorithm for LCS with approximation factor Ω(λ^3).

Truly sublinear time algorithm for LIS with approximation factor Ω(λ^3).

Techniques extend triangle inequality to non-metric settings.

Abstract

Longest common subsequence ($\mathsf{LCS}$) is a classic and central problem in combinatorial optimization. While $\mathsf{LCS}$ admits a quadratic time solution, recent evidence suggests that solving the problem may be impossible in truly subquadratic time. A special case of $\mathsf{LCS}$ wherein each character appears at most once in every string is equivalent to the longest increasing subsequence problem ($\mathsf{LIS}$) which can be solved in quasilinear time. In this work, we present novel algorithms for approximating $\mathsf{LCS}$ in truly subquadratic time and $\mathsf{LIS}$ in truly sublinear time. Our approximation factors depend on the ratio of the optimal solution size over the input size. We denote this ratio by $\lambda$ and obtain the following results for $\mathsf{LCS}$ and $\mathsf{LIS}$ without any prior knowledge of $\lambda$. $\bullet$ A truly subquadratic time algorithm for $\mathsf{LCS}$ with approximation factor $\Omega(\lambda^3)$. $\bullet$A truly sublinear time algorithm for $\mathsf{LIS}$ with approximation factor $\Omega(\lambda^3)$. Triangle inequality was recently used by [Boroujeni, Ehsani, Ghodsi, HajiAghayi and Seddighin SODA 2018] and [Charkraborty, Das, Goldenberg, Koucky and Saks FOCS 2018] to present new approximation algorithms for edit distance. Our techniques for $\mathsf{LCS}$ extend the notion of triangle inequality to non-metric settings.