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Longest common subsequence (LCS) is one of the most fundamental problems incombinatorial optimization. Apart from theoretical importance, LCS has enormousapplications in bioinformatics, revision control systems, and data comparisonprograms. Although a simple dynamic program computes LCS in quadratic time, ithas been recently proven that the problem admits a conditional lower bound andmay not be solved in truly subquadratic time. In addition to this, LCS isnotoriously hard with respect to approximation algorithms. Apart from a trivialsampling 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 itsdual problem edit distance for which several linear time solutions are obtainedin the past two decades.

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