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. 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. Whether the RNA folding problem can be solved in 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. However, their proof requires alphabet size >= 36 to work, making the result biologically irrelevant. In this paper, by constructing the gadgets using a lemma of Bringmann and Künnemann (FOCS'15) and surrounding them with some carefully designed sequences, we improve upon the framework of Abboud et al. to handle the case of alphabet size 4, yielding a conditional lower bound for the RNA folding problem. We also investigate the Dyck edit distance problem. We demonstrate a reduction from RNA folding problem to Dyck edit distance problem of alphabet size 10, establishing a connection between the two fundamental string problems. This leads to a much simpler proof of the conditional lower bound for Dyck edit distance problem given by Abboud et al. and lowers the required alphabet size for the lower bound to work.