Maximum Agreement and Compatible Supertrees

Abstract

Given a set of leaf-labelled trees with identical leaf sets, the MAST problem, respectively MCT problem, consists of finding a largest subset of leaves such that all input trees restricted to these leaves are isomorphic, respectively compatible. In this paper, we propose extensions of these problems to the context of supertree inference, where input trees have non-identical leaf sets. This situation is of particular interest in phylogenetics. The resulting problems are called SMAST and SMCT.A sufficient condition is given that identifies cases where these problems can be solved by resorting to MAST and MCT as subproblems. This condition is met, for instance, when only two input trees are considered. Then we give algorithms for SMAST and SMCT that benefit from the link with the subtree problems. These algorithms run in time linear to the time needed to solve MAST, respectively MCT, on an instance of the same or smaller size.It is shown that arbitrary instances of SMAST and SMCT can be turned in polynomial time into instances composed of trees with a bounded number of leaves.SMAST is shown to be W[2]-hard when the considered parameter is the number of input leaves that have to be removed to obtain the agreement of the input trees. A simlar result holds for SMCT. Moreover, the corresponding optimization problems, that is the complements of SMAST and SMCT, can not be approximated in polynomial time within a constant factor, unless P = NP. These results also hold when the input trees have a bounded number of leaves.The presented results apply to both collections of rooted and unrooted trees.

Linear time 3-approximation for the MAST problem

“…By requiring that induced subtrees of the input trees are compatible, and not strictly isomorphic, MCT usually leads to selecting a larger set of leaves than allowed by MAST. Note that another variant of MAST has been recently proposed to build phylogenetic supertrees, where input trees have different leaf sets [Berry and Nicolas 2004].…”

“…linear time, significantly improving on the former O(kn 4 ) time algorithm of [Amir and Keselman 1997], refined into an O(kn 3 ) algorithm in [Berry and Nicolas 2004]. The improvement in the complexity results from ordering the subtrees of two compared trees such that conflicting triples of leaves are readily identified from the minimum and maximum leaves contained in a subtree.…”

Hierarchical Clustering Using Constraints

“…Following the notation and terminology of [7], a tree T has a leaf set L(T ) in bijection with a label set. Each internal node (including the root) has at least two children.…”

“…Consider the trees T 1 , T 2 and T 3 ( Figure 5) where L(T 1 )={1, 2, 3, 4, 5, 6, 11, 12, 13, 14, 15}, L(T 2 ) = {1, 2,3,4,5,6,7,8,9, 10}, L(T 3 ) = {7, 8, 9, 10, 20, 21, 22, 23, 24}, where…”

“…Semple and Steel [6] proposed MinCutSupertree algorithm to build a rooted supertree for a family of rooted weighted binary trees. Berry and Nicholas [7] propose MergeTrees, an algorithm for constructing a supertree compatible with two binary trees by grafting "specific subtrees" or "specific leaves" of one tree onto the other. For |T | ≥ 3, they apply MergeTrees repeatedly to pairs of trees, each time reducing |T | by 1.…”