Settling the Intractability of Multiple Alignment

In this paper some of the most fundamental problems in computational biology are proved intractable. The following problems are shown NP-hard for all binary or larger alphabets under all xed metrics: Multiple Alignment with SP-score, Star Alignment, and Tree Alignment (for a given phylogeny) . Earlier these problems have only been shown intractable for sporadic alphabets and distances, here the intractability is settled. Moreover, Consensus Patterns and Substring Parsimony are shown NP-hard