Maximum Stacking Base Pairs: Hardness and Approximation by Nonlinear Linear Programming-Rounding.

Maximum stacking base pairs is a fundamental combinatorial problem from ribonucleic acid (RNA) secondary structure prediction under the energy model. The basic maximum stacking base pairs problem can be described as: given an RNA sequence, find a maximum number of base pairs such that each chosen base pair has at least one parallel and adjacent partner (i.e., they form a stacking). This problem is NP-hard, no matter whether the candidate base pairs follow the biology principle or are given explicitly as input. This article investigates a restricted version of this problem where the base pairs are given as input and each base is associated with at most k (a constant) base pairs. We show that this restricted version is still APX-hard, even if the base pairs are weighted. Moreover, by a nonlinear LP-rounding method, we present an approximation algorithm with a factor 32(k-1)3e38(k-1)e-1. Applying our algorithms on the simulated data, the actual approximation factor is in fact much better than this theoretical bound.