Complexity of Pseudoknot Prediction in Simple Models

Efficient exact algorithms for finding optimal secondary structures of RNA sequences have been known for a quarter of a century. However, these algorithms are restricted to structures without overlapping base pairs, or pseudoknots. The ability to include pseudoknots has gained increased attention over the last five years, but three recent publications indicate that this might leave the problem intractable. In this paper we further investigate the complexity of the pseudoknot prediction problem in two simple models based on base pair stacking. We confirm the intractability of pseudoknot prediction by proving it hard for binary strings in one model, and for strings over an unbounded alphabet in the other model. Conversely, we are also able to present a polynomial time algorithm for pseudoknot prediction for strings over a fixed size alphabet in the second model and a polynomial time approximation scheme for pseudoknot prediction for strings over a fixed size alphabet in the first model.