# Improved approximation algorithms for tree alignment

## Abstract

Multiple sequence alignment is a task at the heart of much of current computational biology [4]. Several different objective functions have been proposed to formalize the task of multiple sequence alignment, but efficient algorithms are lacking in each case. Thus multiple sequence alignment is one of the most critical, essentially unsolved problems in computational biology. In this paper we consider one of the more compelling objective functions for multiple sequence alignment, formalized as the *tree alignment problem*. Previously in [15], a factor-of-two approximation method was developed for tree alignment, which ran in *cubic* time (as a function of the number of fixed length strings to be aligned), along with a polynomial time approximation scheme (PTAS) for the problem. However, the PTAS in [15] had a running time which made it impractical to reduce the error bound much below two for small size biological sequences (100 characters long).

In this paper we first develop a factor-of-two approximation algorithm which runs in *quadratic* time, and then use it to develop a PTAS which has a smaller guaranteed error bound and a vastly improved worst case running time compared to the scheme in [15]. With the new approximation scheme, it is now practical to guarantee an error bound of 1.583 for strings of lengths 200 characters or less.

## Key words

tree alignment approximation algorithm and polynomial time approximation scheme## Preview

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