Hardness of RNA Folding Problem with Four Symbols

An RNA sequence is a string composed of four types of nucleotides, $A, C, G$, and $U$. The goal of the RNA folding problem is to find a maximum cardinality set of crossing-free pairs of the form ${A,U}$ or ${C,G}$ in a given RNA sequence. The problem is central in bioinformatics and has received much attention over the years. Abboud, Backurs, and Williams (FOCS 2015) demonstrated a conditional lower bound for a generalized version of the RNA folding problem based on a conjectured hardness of the $k$-clique problem. Their lower bound requires the RNA sequence to have at least 36 types of symbols, making the result not applicable to the RNA folding problem in real life (i.e., alphabet size 4). In this paper, we present an improved lower bound that works for the alphabet size 4 case. We also investigate the Dyck edit distance problem, which is a string problem closely related to RNA folding. We demonstrate a reduction from RNA folding to Dyck edit distance with alphabet size 10. This leads to a much simpler proof of the conditional lower bound for Dyck edit distance problem given by Abboud, Backurs, and Williams (FOCS 2015), and lowers the alphabet size requirement.