On the Computational Complexity of Gap - 0 Alignment

The gap-0 multiple alignment problem is the problem of nding a multiple alignment of k sequences that minimizes the SP-score for a given scoring matrix M by possibly shifting these sequences but not inserting gaps. It is shown that there is a nonnegative, symmetric scoring matrix M 0 with all diagonal entries equal to zero and a uniform indel penalty that exceeds the largest penalty for character mismatches such that the gap-0 multiple alignment problem with respect to M 0 is MAX-SNP-hard in the number of sequences. It is also shown that there is a scoring matrix M 1 with all these properties that in addition satisses the triangle inequality such that the gap-0 multiple alignment problem with respect to M 1 is NP-hard in the number of sequences.