Improved Approximation Algorithms for Dyck Edit Distance and RNA Folding

The Dyck language, which consists of well-balanced sequences of parentheses, is one of the most fundamental context-free languages. The Dyck edit distance quantifies the number of edits (character insertions, deletions, and substitutions) required to make a given parenthesis sequence well-balanced. RNA Folding involves a similar problem, where a closing parenthesis can match an opening parenthesis of the same type irrespective of their ordering. For example, in RNA Folding, both $\tt{()}$ and $\tt{)(}$ are valid matches, whereas the Dyck language only allows $\tt{()}$ as a match. Using fast matrix multiplication, it is possible to compute their exact solutions of both problems in time $O(n^{2.824})$. Whereas combinatorial algorithms would be more desirable, the two problems are known to be at least as hard as Boolean matrix multiplication. In terms of fast approximation algorithms that are combinatorial in nature, both problems admit an $\epsilon n$-additive approximation in $\tilde{O}(\frac{n^2}{\epsilon})$ time. Further, there is a $O(\log n)$-factor approximation algorithm for Dyck edit distance in near-linear time. In this paper, we design a constant-factor approximation algorithm for Dyck edit distance that runs in $O(n^{1.971})$ time. Moreover, we develop a $(1+\epsilon)$-factor approximation algorithm running in $\tilde{O}(\frac{n^2}{\epsilon})$ time, which improves upon the earlier additive approximation. Finally, we design a $(3+\epsilon)$-approximation that takes $\tilde{O}(\frac{nd}{\epsilon})$ time, where $d\ge 1$ is an upper bound on the sought distance. As for RNA folding, for any $s\ge1$, we design a factor-$s$ approximation algorithm that runs in $O(n+(\frac{n}{s})^3)$ time. To the best of our knowledge, this is the first nontrivial approximation algorithm for RNA Folding that can go below the $n^2$ barrier. All our algorithms are combinatorial.