Polynomial algorithms for protein similarity search for restricted mRNA structures

In this paper we consider the problem of computing an mRNA sequence of maximal similarity for a given mRNA of secondary structure constraints, introduced by Backofen et al. in [R. Backofen, N.S. Narayanaswamy, F. Swidan, On the complexity of protein similarity search under mRNA structure constraints, in: Proceedings of the Annual Symposium of Theoretical Aspects of Computer Science, in: LNCS, vol. 2285, Springer, 2002, pp. 274-286] denoted as the MRSO problem. The problem is known to be NP-complete for planar associated implied structure graphs of vertex degree at most 3. In [G. Blin, G. Fertin, D. Hermelin, S. Vialette, Fixed-parameter algorithms for protein similarity search under mRNA structure constraints, in: Proceedings of Graph-Theoretical Concepts in Computer Science, in: LNCS, vol. 3787, Springer, 2005, pp. 271-282] a first polynomial dynamic programming algorithms for MRSO on implied structure graphs with maximum vertex degree 3 of bounded cut-width is shown. We give a simple but much more general polynomial dynamic programming solution for the MRSO problem for associated implied structure graphs of bounded clique-width. Our result implies that MRSO is polynomial for graphs of bounded tree-width, co-graphs, P-4-sparse graphs, and distance hereditary graphs. Further we conclude that the problem of comparing two solutions for MRSO is hard for the class p(parallel to)(NP), which is defined as the set of problems which can be solved in polynomial time with a number of parallel queries to an oracle in (c) 2007 Elsevier B.V. All rights reserved.