Approximating the complement of the maximum compatible subset of leaves of k trees

Abstract

We address a combinatorialprobl em which arises in computationalph ylogenetics. In this problem we are given a set of unrooted (not necessarily binary) trees each leaf-labelled by the same set S, and we wish to remove a minimum number of leaves so that the resultant trees share a common refinement (i.e. they are “compatible”. If we assume the input trees are all binary, then this is simply the Maximum Agreement Subtree problem (MAST), for which much is already known. However, if the input trees need not be binary, then the problem is much more computationally intensive: it is NP-hard for just two trees, and solvable in polynomial time for any number k of trees when all trees have bounded degree. In this paper we present an O(k2n2) 4-approximation algorithm and an O(k2n3) 3-approximation algorithm for the general case of this problem.