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

An edge-weighted tree is called ultrametric if the distances from the root to all the leaves in the tree are equal. For an n by n distance matrix M, t he minimum ultrametric tree for M is an ultrametric tree T = (V, E, w) with leaf set {1, ..., n} such that d(T)(i, j) greater than or equal to M[i, j] for all i, j and Sigma(e is an element of E)w(e) is minimum, where d(T)(i, j) is the distance between i and j on T. Constructing minimum ultrametric trees from distance matrices is an important problem in computational biolo gy. In this paper, we examine its computational complexity and approximabil ity. When the distances satisfy the triangle inequality, we show that the m inimum 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 de velop an efficient branch-and-bound algorithm for constructing the minimum ultrametric tree for both metric and non-metric inputs. The experimental re sults show that it can find an optimal solution for 25 species within reaso nable time, while, to the best of our knowledge, there is no report of algo rithms solving the problem even for 12 species.