Alignment Between Two Multiple Alignments

Alignment of two multiple alignments arises naturally when constructing approximate multiple sequence alignments progressively. In this paper, we consider the problem of alignment of two multiple alignments with SP-score and linear gap costs.When there is no gap opening cost, this problem can be solved using the well-known dynamic programming algorithm for two sequences by viewing each column in the multiple alignments as an element. However if there are gap opening costs (sometimes referred as affine gap costs) then the problem becomes non-trivial. Gotoh [4] suggested a procedure for this problem and stated that "the total arithmetic operations used is close to (quadratic) in typical cases". Kececioglu and Zhang [7] gave heuristic algorithms based on optimistic and pessimistic gap counts and conjectured that this problem is NP-complete. In this paper we prove that this problem is indeed NP-complete and therefore settle this open problem: We then propose another heuristic algorithm for this problem.