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CYCLE KILLER... QU’ESTCE QUE C’EST? ON THE COMPARATIVE APPROXIMABILITY OF HYBRIDIZATION NUMBER AND DIRECTED FEEDBACK VERTEX SET
"... We show that the problem of computing the hybridization number of two rooted binary phylogenetic trees on the same set of taxa X has a constant factor polynomialtime approximation if and only if the problem of computing a minimumsize feedback vertex set in a directed graph (DFVS) has a constant f ..."
Abstract

Cited by 11 (7 self)
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We show that the problem of computing the hybridization number of two rooted binary phylogenetic trees on the same set of taxa X has a constant factor polynomialtime approximation if and only if the problem of computing a minimumsize feedback vertex set in a directed graph (DFVS) has a constant factor polynomialtime approximation. The latter problem, which asks for a minimum number of vertices to be removed from a directed graph to transform it into a directed acyclic graph, is one of the problems in Karp’s seminal 1972 list of 21 NPcomplete problems. Despite considerable attention from the combinatorial optimization community, it remains to this day unknown whether a constant factor polynomialtime approximation exists for DFVS. Our result thus places the (in)approximability of hybridization number in a much broader complexity context, and as a consequence we obtain that it inherits inapproximability results from the problem Vertex Cover. On the positive side, we use results from the DFVS literature to give an O(log r log log r) approximation for the hybridization number where r is the correct value.
A simple fixed parameter tractable algorithm for computing the hybridization number of two (not necessarily binary) trees
, 2012
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A reconstruction problem for a class of phylogenetic networks with lateral gene transfers
 ALGORITHMS MOL BIOL (2015) 10:28
, 2015
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Hybridization Number on Three Trees
, 2014
"... Phylogenetic networks are leaflabelled directed acyclic graphs that are used to describe nontreelike evolutionary histories and are thus a generalization of phylogenetic trees. The hybridization number of a phylogenetic network is the sum of all indegrees minus the number of nodes plus one. The Hy ..."
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Phylogenetic networks are leaflabelled directed acyclic graphs that are used to describe nontreelike evolutionary histories and are thus a generalization of phylogenetic trees. The hybridization number of a phylogenetic network is the sum of all indegrees minus the number of nodes plus one. The Hybridization Number problem takes as input a collection of phylogenetic trees and asks to construct a phylogenetic network that contains an embedding of each of the input trees and has a smallest possible hybridization number. We present an algorithm for the Hybridization Number problem on three binary phylogenetic trees on n leaves, which runs in time O(ckpoly(n)), with k the hybridization number of an optimal network and c some positive constant. For the case of two trees, an algorithm with running time O(3.18kn) was proposed before whereas an algorithm with running time O(ckpoly(n)) for more than two trees had prior to this article remained elusive. The algorithm for two trees uses the close connection to acyclic agreement forests to achieve a linear exponent in the running time, while previous algorithms for more than two trees (explicitly or implicitly) relied on a brute force search through all possible underlying network topologies, leading to running times that are not O(ckpoly(n)), for any c. The connection to acyclic agreement forests is much weaker for
ON COMPUTING THE MAXIMUM PARSIMONY SCORE OF A PHYLOGENETIC NETWORK
"... Abstract. Phylogenetic networks are used to display the relationship of different species whose evolution is not treelike, which is the case, for instance, in the presence of hybridization events or horizontal gene transfers. Tree inference methods such as Maximum Parsimony need to be modified in or ..."
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Abstract. Phylogenetic networks are used to display the relationship of different species whose evolution is not treelike, which is the case, for instance, in the presence of hybridization events or horizontal gene transfers. Tree inference methods such as Maximum Parsimony need to be modified in order to be applicable to networks. In this paper, we discuss two different definitions of Maximum Parsimony on networks, “hardwired ” and “softwired”, and examine the complexity of computing them given a network topology and a character. By exploiting a link with the problem Multicut, we show that computing the hardwired parsimony score for 2state characters is polynomialtime solvable, while for characters with more states this problem becomes NPhard but is still approximable and fixed parameter tractable in the parsimony score. On the other hand we show that, for the softwired definition, obtaining even weak approximation guarantees is already difficult for binary characters and restricted network topologies, and fixedparameter tractable algorithms in the parsimony score are unlikely. On the positive side we show that computing the softwired parsimony score is fixedparameter tractable in the level of the network, a natural parameter describing how tangled reticulate activity is in the network. Finally, we show that both the hardwired and softwired parsimony score can be computed efficiently using Integer Linear Programming. The software has been made freely available. 1.
A note on efficient computation of hybridization number via softwired clusters
"... Abstract. Here we present a new fixed parameter tractable algorithm to compute the hybridization number r of two rooted binary phylogenetic trees on taxon set X in time (6r)r · poly(n), where n = X . The novelty of this approach is that it avoids the use of Maximum Acyclic Agreement Forests (MAAFs ..."
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Abstract. Here we present a new fixed parameter tractable algorithm to compute the hybridization number r of two rooted binary phylogenetic trees on taxon set X in time (6r)r · poly(n), where n = X . The novelty of this approach is that it avoids the use of Maximum Acyclic Agreement Forests (MAAFs) and instead exploits the equivalence of the problem with a related problem from the softwired clusters literature. This offers an alternative perspective on the underlying combinatorial structure of the hybridization number problem. 1