Conditional Lower Bound for RNA Folding Problem

An RNA sequence is a string composed of four types of nucleotides, $A, C, G$, and $U$. Given an RNA sequence, the goal of the RNA folding problem is to find a maximum cardinality set of crossing-free pairs of the form $\{A,U\}$ or $\{C,G\}$. The problem is central in bioinformatics and has received much attention over the years. However, the current best algorithm for the problem still takes $\mathcal{O}\left(\frac{n^3}{\log^2 (n)}\right)$ time, which is only a slight improvement over the classic $\mathcal{O}(n^3)$ dynamic programming algorithm. Whether the RNA folding problem can be solved in $\mathcal{O}(n^{3-\epsilon})$ time remains an open problem. Recently, Abboud, Backurs, and Williams (FOCS'15) made the first progress by showing a conditional lower bound for a generalized version of the RNA folding problem based on a conjectured hardness of the $k$-clique problem. A drawback of their work is that they require the RNA sequence to have at least 36 types of letters, making their result biologically irrelevant. In this paper, we show that by constructing the gadgets using a powerful lemma of Bringmann and K\"unnemann (FOCS'15) and surrounding them with some carefully designed sequences, the framework of Abboud et al. can be improved upon to work for the case where the alphabet size is 4, yielding a conditional lower bound for the RNA folding problem.