Abstract. Given a set of leaf-labelled trees with identical leaf sets, the well-known MAST problem consists of finding a subtree homeomorphically included in all input trees and with the largest number of leaves. MAST and its variant called MCT are of particular interest in computational biology. This paper presents positive and negative results on the approximation of MAST, MCT and their complement versions, denoted CMAST and CMCT. For CMAST and CMCT on rooted trees we give 3-approximation algorithms achieving significantly lower running times than those previously known. In particular, the algorithm for CMAST runs in linear time. The approximation threshold for CMAST, resp. CMCT, is shown to be the same whenever collections of rooted trees or of unrooted trees are considered. Moreover, hardness of approximation results are stated for CMAST, CMCT and MCT on small number of trees, and for MCT on unbounded number of trees.

Abstract. Given a set N of phylogenetic networks, the maximum agreement phylogenetic subnetwork problem (MASN) asks for a subnetwork embedded in every Ni ∈ N with as many leaves as possible. MASN can be used to identify shared branching structure among phylogenetic networks or to measure their similarity. In this chapter, we prove that the general case of MASN is NP-hard already for two phylogenetic networks (in fact, even if one of the two input networks is a binary tree), but that the problem can be solved efficiently if each of the two input phylogenetic networks exhibits a nested structure. For this purpose, we introduce the concept of a nested phylogenetic network and study some of the underlying fundamental combinatorial properties. We first show that the total number of nodes |V (N) | in any nested phylogenetic network N with n leaves and nesting depth d is O(n(d + 1)). We then describe a simple algorithm for testing if a given phylogenetic network is nested, and if so, determining its nesting depth in O(|V (N) | · (d + 1)) time. Next, we present a polynomial-time algorithm for MASN for two nested phylogenetic networks N1, N2. Its running time is O(|V (N1) | · |V (N2) | · (d1 + 1) · (d2 + 1)), where d1 and d2 denote the nesting depths of N1 and N2, respectively. In contrast, the previously fastest algorithm for this problem runs in O(|V (N1) | · |V (N2) | · 2 f1+f2) time, where f1 ≥ d1 and f2 ≥ d2. Finally, we prove that if the nodes are allowed to have outdegree greater than 2 then the problem becomes NP-hard even if restricted to two phylogenetic networks with nesting depth 1. 1

Given a set of evolutionary trees on a same set of taxa, the maximum agreement subtree problem (MAST), respectively maximum compatible tree problem (MCT), consists of finding a largest subset of taxa such that all input trees restricted to these taxa are isomorphic, respectively compatible. These problems

Given a set of leaf-labeled trees with identical leaf sets, the well-known Maximum Agreement SubTree problem (MAST) consists of finding a subtree homeomorphically included in all input trees and with the largest number of leaves. Its variant called Maximum Compatible Tree (MCT) is less stringent, as it allows the input trees to be refined. Both problems are of particular interest in computational biology, where trees encountered have often small degrees. In this paper, we study the parameterized complexity of MAST and MCT with respect to the maximum degree, denoted by D, of the input trees. Although MAST is polynomial for bounded D [1, 6, 3], we show that the problem is W[1]-hard with respect to parameter D. Moreover, relying on recent advances in parameterized complexity we obtain a tight lower bound: while MAST can be solved in O(N O(D)) time where N denotes the input length, we show that an O(N o(D) ) bound is not achievable, unless SNP ⊆ SE. We also show that MCT is W[1]-hard with respect to D, and that MCT cannot be solved in O(N o(2D/2)) time, unless SNP ⊆ SE. 1

We propose and study the Maximum Constrained Agreement Sub-tree (MCAST) problem, which is a variant of the classical Maximum Agreement Subtree (MAST) problem. Our problem allows users to ap-ply their domain knowledge to control the construction of the agreement subtrees in order to get better results. We show that the MCAST prob-lem can be reduced to the MAST problem in linear time and thus we have algorithms for MCAST with running times matching the fastest known algorithms for MAST.

Consider a set of labels L and a set of trees T = {T (1) , T (2) ,..., T (k) } where each tree T (i) is distinctly leaf-labeled by some subset of L. One fundamental problem is to find the biggest tree (denoted as supertree) to represent T which minimizes the disagreements with the trees in T under certain criteria. This problem finds applications in phylogenetics, database, and data mining. In this paper, we focus on two particular supertree problems, namely, the maximum agreement supertree problem (MASP) and the maximum compatible supertree problem (MCSP). These two problems are known to be NP-hard for k ≥ 3. This paper gives the first polynomial time algorithms for both MASP and MCSP when both k and the maximum degree D of the trees are constant.