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33
Constructing the simplest possible phylogenetic network from triplets,”
 Algorithmica,
, 2011
"... Abstract A phylogenetic network is a directed acyclic graph that visualizes an evolutionary history containing socalled reticulations such as recombinations, hybridizations or lateral gene transfers. Here we consider the construction of a simplest possible phylogenetic network consistent with an i ..."
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Cited by 20 (5 self)
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Abstract A phylogenetic network is a directed acyclic graph that visualizes an evolutionary history containing socalled reticulations such as recombinations, hybridizations or lateral gene transfers. Here we consider the construction of a simplest possible phylogenetic network consistent with an input set T , where T contains at least one phylogenetic tree on three leaves (a triplet) for each combination of three taxa. To quantify the complexity of a network we consider both the total number of reticulations and the number of reticulations per biconnected component, called the level of the network. We give polynomialtime algorithms for constructing a level1 respectively a level2 network that contains a minimum number of reticulations and is consistent with T (if such a network exists). In addition, we show that if T is precisely equal to the set of triplets consistent with some network, then we can construct such a network with smallest possible level in time O(T  k+1 ), if k is a fixed upper bound on the level of the network.
Uniqueness, intractability and exact algorithms: reflections on levelk phylogenetic networks
, 2009
"... Phylogenetic networks provide a way to describe and visualize evolutionary histories that have undergone socalled reticulate evolutionary events such as recombination, hybridization or horizontal gene transfer. The level k of a network determines how nontreelike the evolution can be, with level0 ..."
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Cited by 16 (6 self)
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Phylogenetic networks provide a way to describe and visualize evolutionary histories that have undergone socalled reticulate evolutionary events such as recombination, hybridization or horizontal gene transfer. The level k of a network determines how nontreelike the evolution can be, with level0 networks being trees. We study the problem of constructing levelk phylogenetic networks from triplets, i.e. phylogenetic trees for three leaves (taxa). We give, for each k, a levelk network that is uniquely defined by its triplets. We demonstrate the applicability of this result by using it to prove that (1) for all k ≥ 1it is NPhard to construct a levelk network consistent with all input triplets, and (2) for all k ≥ 0 it is NPhard to construct a levelk network consistent with a maximum number of input triplets, even when the input is dense. As a response to this intractability, we give an exact algorithm for constructing level1 networks consistent with a maximum number of input triplets.
Worstcase optimal approximation algorithms for maximizing triplet consistency within phylogenetic networks
, 2008
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New Results on Optimizing Rooted Triplets Consistency
"... Abstract. A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomialtime approximability of two related optimization problems called the maximum rooted triplets consistency problem (MaxRTC) and ..."
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Abstract. A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomialtime approximability of two related optimization problems called the maximum rooted triplets consistency problem (MaxRTC) and the minimum rooted triplets inconsistency problem (MinRTI) in which the input is a set R of rooted triplets, and where the objectives are to find a largest cardinality subset of R which is consistent and a smallest cardinality subset of R whose removal from R results in a consistent set, respectively. We first show that a simple modification to Wu’s BestPairMergeFirst heuristic [25] results in a bottomupbased 3approximation for MaxRTC. We then demonstrate how any approximation algorithm for MinRTI could be used to approximate MaxRTC, and thus obtain the first polynomialtime approximation algorithm for MaxRTC with approximation ratio smaller than 3. Next, we prove that for a set of rooted triplets generated under a uniform random model, the maximum fraction of triplets which can be consistent with any tree is approximately one third, and then provide a deterministic construction of a triplet set having a similar property which is subsequently used to prove that both MaxRTC and MinRTI are NPhard even if restricted to minimally dense instances. Finally, we prove that MinRTI cannot be approximated within a ratio of Ω(log n) in polynomial time, unless P = NP. 1
Close lower and upper bounds for the minimum reticulate network of multiple phylogenetic trees
 Bioinformatics [ISMB
"... Motivation: Reticulate network is a model for displaying and quantifying the effects of complex reticulate processes on the evolutionary history of species undergoing reticulate evolution. A central computational problem on reticulate networks is: given a set of phylogenetic trees (each for some reg ..."
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Cited by 11 (1 self)
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Motivation: Reticulate network is a model for displaying and quantifying the effects of complex reticulate processes on the evolutionary history of species undergoing reticulate evolution. A central computational problem on reticulate networks is: given a set of phylogenetic trees (each for some region of the genomes), reconstruct the most parsimonious reticulate network (called the minimum reticulate network) that combines the topological information contained in the given trees. This problem is well known to be NPhard. Thus, existing approaches for this problem either work with only two input trees or make simplifying topological assumptions. Results: We present novel results on the minimum reticulate network problem. Unlike existing approaches, we address the fully general problem: there is no restriction on the number of trees that are input, and there is no restriction on the form of the allowed reticulate network. We present lower and upper bounds on the minimum number of reticulation events in the minimum reticulate network (and infer an approximately parsimonious reticulate network). A program called PIRN implements these methods, which also outputs a graphical representation of the inferred network. Empirical results on simulated and biological data show that our methods are practical for a wide range of data. More importantly, the lower and upper bounds match for many datasets (especially when the number of trees is small or reticulation level is low), and this allows us to solve the minimum reticulate network problem exactly for these datasets. Availability: A software tool, PIRN, is available for download from the web page:
On the elusiveness of clusters
"... Abstract—Rooted phylogenetic networks are often used to represent conflicting phylogenetic signals. Given a set of clusters, a network is said to represent these clusters in the softwired sense if, for each cluster in the input set, at least one tree embedded in the network contains that cluster. Mo ..."
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Abstract—Rooted phylogenetic networks are often used to represent conflicting phylogenetic signals. Given a set of clusters, a network is said to represent these clusters in the softwired sense if, for each cluster in the input set, at least one tree embedded in the network contains that cluster. Motivated by parsimony we might wish to construct such a network using as few reticulations as possible, or minimizing the level of the network, i.e. the maximum number of reticulations used in any “tangled ” region of the network. Although these are NPhard problems, here we prove that, for every fixedk ≥ 0, it is polynomialtime solvable to construct a phylogenetic network with level equal to k representing a cluster set, or to determine that no such network exists. However, this algorithm does not lend itself to a practical implementation. We also prove that the comparatively efficient CASS algorithm correctly solves this problem (and also minimizes the reticulation number) when input clusters are obtained from two not necessarily binary gene trees on the same set of taxa but does not always minimize level for general cluster sets. Finally, we describe a new algorithm which generates in polynomialtime all binary phylogenetic networks with exactly r reticulations representing a set of input clusters (for every fixed r ≥ 0).
LEVELK PHYLOGENETIC NETWORK CAN BE CONSTRUCTED FROM A DENSE TRIPLET SET IN POLYNOMIAL TIME
, 2009
"... Given a dense triplet set T, there arise two interesting questions [7]: Does there exists any phylogenetic network consistent with T? And if so, can we find an effective algorithm to construct one? For cases of networks of levels k = 0 or 1 or 2, these questions were answered in [1, 6, 7, 8, 10] wit ..."
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Cited by 10 (1 self)
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Given a dense triplet set T, there arise two interesting questions [7]: Does there exists any phylogenetic network consistent with T? And if so, can we find an effective algorithm to construct one? For cases of networks of levels k = 0 or 1 or 2, these questions were answered in [1, 6, 7, 8, 10] with effective polynomial algorithms. For higher levels k, partial answers were recently obtained in [11] with an O(T  k+1) time algorithm for simple networks. In this paper we give a complete answer to the general case, solving a problem of [7]. The main idea is to use a special property of SNsets in a levelk network. As a consequence, we can also find the levelk network with the minimum number of reticulations in polynomial time.
Constructing minimal phylogenetic networks from softwired clusters is fixed parameter tractable
, 2011
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