Hardness Results on Local Multiple Alignment of Biological Sequences

This paper studies the local multiple alignment problem, which is, given protein or DNA sequences, to locate a region (i.e., a substring) of fixed length from each sequence so that the score determined from the set of regions is optimized. We consider the following scoring schemes: the relative entropy score (i.e., average information content), the sum-of-pairs score and a relative entropy-like score introduced by Li, et al. We prove that multiple local alignment is NP-hard under each of these scoring schemes. In particular, we prove that multiple local alignment is APX-hard under relative entropy scoring. It implies that unless P =NP there is no polynomial time algorithm whose worst case approximation error can be arbitrarily specified(precisely, a polynomial time approximation scheme). Several related theoretical results are also provided.