Approximating LCS in Linear Time: Beating the $\sqrt{n}$ Barrier

Longest common subsequence (LCS) is one of the most fundamental problems in combinatorial optimization. Apart from theoretical importance, LCS has enormous applications in bioinformatics, revision control systems, and data comparison programs. Although a simple dynamic program computes LCS in quadratic time, it has been recently proven that the problem admits a conditional lower bound and may not be solved in truly subquadratic time. In addition to this, LCS is notoriously hard with respect to approximation algorithms. Apart from a trivial sampling technique that obtains a $n^x$ approximation solution in time $O(n^{2-2x})$ nothing else is known for LCS. This is in sharp contrast to its dual problem edit distance for which several linear time solutions are obtained in the past two decades.