Reducing approximate Longest Common Subsequence to approximate Edit Distance

Given a pair of strings, the problems of computing their Longest Common Subsequence and Edit Distance have been extensively studied for decades. For exact algorithms, LCS and Edit Distance (with character insertions and deletions) are equivalent; the state of the art running time is (almost) quadratic and this is tight under plausible fine-grained complexity assumptions. But for approximation algorithms the picture is different: there is a long line of works with improved approximation factors for Edit Distance, but for LCS (with binary strings) only a trivial $1/2$-approximation was known. In this work we give a reduction from approximate LCS to approximate Edit Distance, yielding the first efficient $(1/2+ε)$-approximation algorithm for LCS for some constant $ε>0$.