Approximation algorithms for optimal RNA secondary structures common to multiple sequences

It is well known that a basic version (i.e., maximizing the number of base-pairs) of the RNA secondary structure prediction problem can be solved in O(n(3)) time by using simple dynamic programming procedures. For this problem, an O(n(3)(log log n)(1/2)/(log n)(1/2)) time exact algorithm and an O(n(2.776) + (1/epsilon)(O(1))) time approximation algorithm which has guaranteed approximation ratio 1 - epsilon for any positive constant epsilon are also known. Moreover, when two RNA sequences are given, there is an O(n(6)) time exact algorithm which can optimize structure and alignments. In this paper, we show an O(n(5)) time approximation algorithm for optimizing structure and alignments of two RNA sequences with assuming that the optimal number of base-pairs is more than O(n(0.75)). We also show that the problem to optimize structure and alignments for given N sequences is NP-hard and introduce a constant-factor approximation algorithm.