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On the approximation of computing evolutionary trees
 in Proceedings of the 11th International Computing and Combinatorics Conference (COCOON’05
, 2005
"... Abstract. Given a set of leaflabelled trees with identical leaf sets, the wellknown 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. Th ..."
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Abstract. Given a set of leaflabelled trees with identical leaf sets, the wellknown 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 3approximation 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.
Efficient algorithms for descendent subtrees comparison of phylogenetic trees with applications to coevolutionary classifications in bacterial genome
 In The 14th Annual International Symposium on Algorithms and Computation (ISAAC’03), Lecture Notes in Computer Science 2906
, 2003
"... Abstract. A phylogenetic tree is a rooted tree with unbounded degree such that each leaf node is uniquely labelled from 1 to n. The descendent subtree of of a phylogenetic tree T is the subtree composed by all edges and nodes of T descending from a vertex. Given a set of phylogenetic trees, we prese ..."
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Abstract. A phylogenetic tree is a rooted tree with unbounded degree such that each leaf node is uniquely labelled from 1 to n. The descendent subtree of of a phylogenetic tree T is the subtree composed by all edges and nodes of T descending from a vertex. Given a set of phylogenetic trees, we present linear time algorithms for finding all leafagree descendent subtrees as well as all isomorphic descendent subtrees. The normalized cluster distance, d(A, B), of two sets is defined by d(A, B) = ∆(A, B)/(A  + B), where ∆(A, B) denotes the symmetric set difference of two sets. We show that computing all pairs normalized cluster distances between descendent subtrees of two phylogenetic trees can be done in O(n 2) time. Since the total size of the outputs will be Θ(n 2), the algorithm is thus computationally optimal. A nearest subtree of a subset of leaves is such a descendent subtree that has the smallest normalized cluster distance to these leaves. Here we show that finding nearest subtrees for a collection of pairwise disjointed subsets of leaves can be done in O(n) time. Several applications of these algorithms in areas of bioinformatics is considered. Among them, we discuss the 2CS (Two component systems) functional analysis and classifications on bacterial genome.
On the approximability of the Maximum Agreement SubTree and Maximum Compatible Tree problems ⋆
, 802
"... The aim of this paper is to give a complete picture of approximability for two tree consensus problems which are of particular interest in computational biology: Maximum Agreement SubTree (MAST) and Maximum Compatible Tree (MCT). Both problems take as input a label set and a collection of trees whos ..."
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The aim of this paper is to give a complete picture of approximability for two tree consensus problems which are of particular interest in computational biology: Maximum Agreement SubTree (MAST) and Maximum Compatible Tree (MCT). Both problems take as input a label set and a collection of trees whose leaf sets are each bijectively labeled with the label set. Define the size of a tree as the number of its leaves. The wellknown MAST problem consists of finding a maximumsized tree that is topologically embedded in each input tree, under labelpreserving embeddings. Its variant MCT is less stringent, as it allows the input trees to be arbitrarily refined. Our results are as follows. We show that MCT is NPhard to approximate within bound n 1−ǫ on rooted trees, where n denotes the size of each input tree; the same approximation lower bound was already known for MAST [1]. Furthermore, we prove that MCT on two rooted trees is not approximable within bound 2 log1−ǫ n unless all problems in NP are solvable in quasipolynomial time; the same result was previously established for MAST on three rooted trees [2] (note that MAST