Maximum agreement subtree in a set of evolutionary trees-metrics and efficient algorithms

In this paper we prove that the maximum homeomorphic agreement subtree problem is 𝒩𝒫-complete for three trees with unbounded degrees. We then show an approximation algorithm of time O(kn⁵) for choosing the species that are not in a maximum agreement subtree of a set of k trees. Our approximation is guaranteed to provide a set that is no more than 4 times the optimum solution. While the set of evolutionary trees may be large in practice, the trees usually have very small degrees, typically no larger than three. We develop a new method for finding a maximum agreement subtree of k trees, of which one has degree bounded by d. This new method enables us to find a maximum agreement subtree in time O(kn^d+1)