Approximation algorithms for tree alignment with a given phylogeny

We study the following fundamental problem in computational molecular biology: Given a set of DNA sequences representing some species and a phylogenetic tree depicting the ancestral relationship among these species, compute an optimal alignment of the sequences by the means of constructing a minimum-cost evolutionary tree. The problem is an important variant of multiple sequence alignment, and is widely known astree alignment. We design an efficient approximation algorithm with performance ratio 2 for tree alignment. The algorithm is then extended to a polynomial-time approximation scheme. The construction actually works for Steiner trees in any metric space, and thus implies a polynomial-time approximation scheme for planar Steiner trees under a given topology (with any constant degree). To our knowledge, this is the first polynomial-time approximation scheme in the fields of computational biology and Steiner trees. The approximation algorithms may be useful in evolutionary genetics practice as they can provide a good initial alignment for the iterative method in [23].