Approximation Algorithms for LCS and LIS with Truly Improved Running Times

Abstract

SIAM Journal on Computing, Ahead of Print.
Abstract. Longest common subsequence (LCS) is a classic and central problem in combinatorial optimization. While LCS admits a quadratic time solution, recent evidence suggests that solving the problem may be impossible in truly subquadratic time. A special case of LCS wherein each character appears at most once in every string is equivalent to the longest increasing subsequence (LIS) problem which can be solved in quasilinear time. In this work, we present novel algorithms for approximating LCS in truly subquadratic time and LIS in truly sublinear time. Our approximation factors depend on the ratio of the optimal solution size to the input size. We denote this ratio by [math] and obtain the following results for LCS and LIS without any prior knowledge of [math]: a truly subquadratic time algorithm for LCS with approximation factor [math] and a truly sublinear time algorithm for LIS with approximation factor [math]. The triangle inequality was recently used by M. Boroujeni, S. Ehsani, M. Ghodsi, M. HajiAghayi, and S. Seddingham [Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, 2018, pp. 1170–1189] and D. Chakraborty, D. Das, E. Goldenberg, M. Koucky, and M. Saks [Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science, 2018, pp. 979–990] to present new approximation algorithms for edit distance. Our techniques for LCS extend the notion of the triangle inequality to nonmetric settings.