Approximation Algorithms for LCS and LIS with Truly Improved Running Times

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.