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...measure (Carillo and Lipman, 1988), all ¡ n¢ 2 scores of the pairwise alignments within the MSA (see De� nition 1.9) are added up. SP methods are obviously de� cient from an evolutionary perspective (=-=Altschul and Lipman, 1989-=-). Consider a tree (Figure 2) constructed for a family containing � ve proteins. The score of a pairwise alignment hA , B i evaluates the probability of evolutionary events on edges (u, A) and (u, B )...

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...k = 2, and a 3k(k−1)3k−2 -approximation algorithm for k ≥ 3. 1 Introduction Computational molecular biology has emerged as one of the most exciting interdisciplinary fields in the past two decades, in part because various biological applications have spawned a large number of interesting combinatorial problems such as multiple sequence alignment [12], sorting by reversals [20], and recently the minimum common partition problem [10]. These problems have attracted considerable attention from computer scientists who took the challenge to design efficient and effective algorithms for solving them [5, 14, 13]. In this paper, we introduce a new combinatorial optimization problem, called the Minimum Common Integer Partition problem (MCIP), which was inspired by our recent work on ortholog assignment and DNA fingerprint assembly. By a partition of a positive integer n we mean a multiset {n1, n2, · · · , nr} of positive integers that add up to exactly n, i.e. ∑r i=1 ni = n, where ni is called a part of n [2, 4]. Given a multiset S = {x1, x2, · · · , xm} of integers with a partition for each integer xi, 1 ≤ i ≤ m, we can define an integer partition of S as the multiset union of these partitions, that i...

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...there are many types of multiple sequence alignment (see (Chan et al. 1994) and references), tree alignment is preferable because it takes evolutionary descent into account in creating the alignment (=-=Altschul and Lipman 1989-=-; Sankoff and Cedergren 1983). The best known algorithms for this problem require time exponential in the length of the longest given sequence (see (Jiang et al. 1994) and references). The NP- and MAX...