  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.

Computational Complexity;Data Structures and Algorithms