Approximation and exact algorithms for constructing minimum ultrametric trees from distance matrices

Abstract

Constructing minimum ultrametric trees from distance matrices is an important problem in computational biology. In this paper, we examine its computational complexity and approximability. When the distances satisfy the triangle inequalities, we show that the minimum ultrametric tree problem can be approximated in polynomial time with error ratio 1.5(1 + ⌈log n⌉), where n is the number of species. We also developed an efficient branch and bound algorithm for constructing the minimum ultrametric tree for both metric and nonmetric inputs. The experimental results show that it can find an optimal solution for 25 species within reasonable time, while, to the best of our knowledge, there is no report of algorithms solving the problem even for 12 species.