  We consider the classic problem of computing the Longest Common Subsequence
(LCS) of two strings of length $n$. While a simple quadratic algorithm has been
known for the problem for more than 40 years, no faster algorithm has been
found despite an extensive effort. The lack of progress on the problem has
recently been explained by Abboud, Backurs, and Vassilevska Williams [FOCS'15]
and Bringmann and K\"unnemann [FOCS'15] who proved that there is no
subquadratic algorithm unless the Strong Exponential Time Hypothesis fails.
This has led the community to look for subquadratic approximation algorithms
for the problem.
  Yet, unlike the edit distance problem for which a constant-factor
approximation in almost-linear time is known, very little progress has been
made on LCS, making it a notoriously difficult problem also in the realm of
approximation. For the general setting, only a naive
$O(n^{\varepsilon/2})$-approximation algorithm with running time
$\tilde{O}(n^{2-\varepsilon})$ has been known, for any constant $0 <
\varepsilon \le 1$. Recently, a breakthrough result by Hajiaghayi, Seddighin,
Seddighin, and Sun [SODA'19] provided a linear-time algorithm that yields a
$O(n^{0.497956})$-approximation in expectation; improving upon the naive
$O(\sqrt{n})$-approximation for the first time.
  In this paper, we provide an algorithm that in time $O(n^{2-\varepsilon})$
computes an $\tilde{O}(n^{2\varepsilon/5})$-approximation with high
probability, for any $0 < \varepsilon \le 1$. Our result (1) gives an
$\tilde{O}(n^{0.4})$-approximation in linear time, improving upon the bound of
Hajiaghayi, Seddighin, Seddighin, and Sun, (2) provides an algorithm whose
approximation scales with any subquadratic running time $O(n^{2-\varepsilon})$,
improving upon the naive bound of $O(n^{\varepsilon/2})$ for any $\varepsilon$,
and (3) instead of only in expectation, succeeds with high probability.
