




Leo van Iersel,
Steven Kelk,
Nela Lekic,
Chris Whidden and
Norbert Zeh. Hybridization Number on Three Rooted Binary Trees is EPT. In SIDMA, Vol. 30(3):16071631, 2016. Keywords: agreement forest, explicit network, FPT, from rooted trees, hybridization, minimum number, phylogenetic network, phylogeny, reconstruction. Note: http://arxiv.org/abs/1402.2136.



Steven Kelk,
Leo van Iersel,
Celine Scornavacca and
Mathias Weller. Phylogenetic incongruence through the lens of Monadic Second Order logic. In JGAA, Vol. 20(2):189215, 2016. Keywords: agreement forest, explicit network, FPT, from rooted trees, hybridization, minimum number, MSOL, phylogenetic network, phylogeny, reconstruction. Note: http://jgaa.info/accepted/2016/KelkIerselScornavaccaWeller2016.20.2.pdf.







Philippe Gambette,
Leo van Iersel,
Steven Kelk,
Fabio Pardi and
Celine Scornavacca. Do branch lengths help to locate a tree in a phylogenetic network? In BMB, Vol. 78(9):17731795, 2016. Keywords: branch length, explicit network, FPT, from network, from rooted trees, NP complete, phylogenetic network, phylogeny, pseudopolynomial, time consistent network, tree containment, tree sibling network. Note: http://arxiv.org/abs/1607.06285.



Leo van Iersel,
Steven Kelk,
Giorgios Stamoulis,
Leen Stougie and
Olivier Boes. On unrooted and rootuncertain variants of several wellknown phylogenetic network problems. 2016. Keywords: explicit network, FPT, from network, from unrooted trees, NP complete, phylogenetic network, phylogeny, reconstruction, tree containment. Note: http://arxiv.org/abs/1609.00544.






Mareike Fischer,
Leo van Iersel,
Steven Kelk and
Celine Scornavacca. On Computing The Maximum Parsimony Score Of A Phylogenetic Network. In SIDMA, Vol. 29(1):559585, 2015. Keywords: APX hard, cluster containment, explicit network, FPT, from network, from sequences, integer linear programming, level k phylogenetic network, NP complete, parsimony, phylogenetic network, phylogeny, polynomial, Program MPNet, reconstruction, software. Note: http://arxiv.org/abs/1302.2430.






Steven Kelk and
Celine Scornavacca. Constructing minimal phylogenetic networks from softwired clusters is fixed parameter tractable. In ALG, Vol. 68(4):886915, 2014. Keywords: explicit network, FPT, from clusters, level k phylogenetic network, phylogenetic network, phylogeny, reconstruction. Note: http://arxiv.org/abs/1108.3653.
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"Here we show that, given a set of clusters C on a set of taxa X, where X=n, it is possible to determine in time f(k)×poly(n) whether there exists a level≤k network (i.e. a network where each biconnected component has reticulation number at most k) that represents all the clusters in C in the softwired sense, and if so to construct such a network. This extends a result from Kelk et al. (in IEEE/ACM Trans. Comput. Biol. Bioinform. 9:517534, 2012) which showed that the problem is polynomialtime solvable for fixed k. By defining "kreticulation generators" analogous to "levelk generators", we then extend this fixed parameter tractability result to the problem where k refers not to the level but to the reticulation number of the whole network. © 2012 Springer Science+Business Media New York."



Leo van Iersel,
Steven Kelk,
Nela Lekic and
Leen Stougie. Approximation algorithms for nonbinary agreement forests. In SIDMA, Vol. 28(1):4966, 2014. Keywords: agreement forest, approximation, from rooted trees, hybridization, minimum number, phylogenetic network, phylogeny, reconstruction. Note: http://arxiv.org/abs/1210.3211.
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"Given two rooted phylogenetic trees on the same set of taxa X, the Maximum Agreement Forest (maf) problem asks to find a forest that is, in a certain sense, common to both trees and has a minimum number of components. The Maximum Acyclic Agreement Forest (maaf) problem has the additional restriction that the components of the forest cannot have conflicting ancestral relations in the input trees. There has been considerable interest in the special cases of these problems in which the input trees are required to be binary. However, in practice, phylogenetic trees are rarely binary, due to uncertainty about the precise order of speciation events. Here, we show that the general, nonbinary version of maf has a polynomialtime 4approximation and a fixedparameter tractable (exact) algorithm that runs in O(4opoly(n)) time, where n = X and k is the number of components of the agreement forest minus one. Moreover, we show that a capproximation algorithm for nonbinary maf and a dapproximation algorithm for the classical problem Directed Feedback Vertex Set (dfvs) can be combined to yield a d(c+3)approximation for nonbinary maaf. The algorithms for maf have been implemented and made publicly available. © 2014 Society for Industrial and Applied Mathematics."



Leo van Iersel and
Steven Kelk. Kernelizations for the hybridization number problem on multiple nonbinary trees. In WG14, Vol. 8747:299311 of LNCS, springer, 2014. Keywords: explicit network, from rooted trees, kernelization, minimum number, phylogenetic network, phylogeny, Program Treeduce, reconstruction. Note: http://arxiv.org/abs/1311.4045.



Leo van Iersel,
Steven Kelk,
Nela Lekic and
Celine Scornavacca. A practical approximation algorithm for solving massive instances of hybridization number for binary and nonbinary trees. In BMCB, Vol. 15(127):112, 2014. Keywords: agreement forest, approximation, explicit network, from rooted trees, phylogenetic network, phylogeny, Program CycleKiller, Program TerminusEst, reconstruction. Note: http://dx.doi.org/10.1186/1471210515127.








Teresa Piovesan and
Steven Kelk. A simple fixed parameter tractable algorithm for computing the hybridization number of two (not necessarily binary) trees. In TCBB, Vol. 10(1):1825, 2013. Keywords: FPT, from rooted trees, phylogenetic network, phylogeny, Program TerminusEst, reconstruction. Note: http://arxiv.org/abs/1207.6090.
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"Here, we present a new fixed parameter tractable algorithm to compute the hybridization number (r) of two rooted, not necessarily binary phylogenetic trees on taxon set (X) in time ((6r r) · poly(n)), where (n= X). The novelty of this approach is its use of terminals, which are maximal elements of a natural partial order on (X), and several insights from the softwired clusters literature. This yields a surprisingly simple and practical boundedsearch algorithm and offers an alternative perspective on the underlying combinatorial structure of the hybridization number problem. © 20042012 IEEE."



Eric Bapteste,
Leo van Iersel,
Axel Janke,
Scott Kelchner,
Steven Kelk,
James O. McInerney,
David A. Morrison,
Luay Nakhleh,
Mike Steel,
Leen Stougie and
James B. Whitfield. Networks: expanding evolutionary thinking. In Trends in Genetics, Vol. 29(8):439441, 2013. Keywords: abstract network, explicit network, phylogenetic network, phylogeny, reconstruction. Note: http://bioinf.nuim.ie/wpcontent/uploads/2013/06/BaptesteTiG2013.pdf.
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"Networks allow the investigation of evolutionary relationships that do not fit a tree model. They are becoming a leading tool for describing the evolutionary relationships between organisms, given the comparative complexities among genomes. © 2013 Elsevier Ltd."










Steven Kelk,
Celine Scornavacca and
Leo van Iersel. On the elusiveness of clusters. In TCBB, Vol. 9(2):517534, 2012. Keywords: explicit network, from clusters, from rooted trees, from triplets, level k phylogenetic network, phylogenetic network, phylogeny, Program Clustistic, reconstruction, software. Note: http://arxiv.org/abs/1103.1834.



Steven Kelk,
Leo van Iersel,
Nela Lekic,
Simone Linz,
Celine Scornavacca and
Leen Stougie. Cycle killer... qu'estce que c'est? On the comparative approximability of hybridization number and directed feedback vertex set. In SIDMA, Vol. 26(4):16351656, 2012. Keywords: agreement forest, approximation, explicit network, from rooted trees, minimum number, phylogenetic network, phylogeny, Program CycleKiller, reconstruction. Note: http://arxiv.org/abs/1112.5359, about the title.
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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. Copyright © by SIAM."





Leo van Iersel,
Steven Kelk,
Nela Lekic and
Celine Scornavacca. A practical approximation algorithm for solving massive instances of hybridization number. In WABI12, Vol. 7534(430440) of LNCS, springer, 2012. Keywords: agreement forest, approximation, explicit network, from rooted trees, hybridization, phylogenetic network, phylogeny, Program CycleKiller, Program Dendroscope, Program HybridNET, reconstruction, software. Note: http://arxiv.org/abs/1205.3417.
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"Reticulate events play an important role in determining evolutionary relationships. The problem of computing the minimum number of such events to explain discordance between two phylogenetic trees is a hard computational problem. In practice, exact solvers struggle to solve instances with reticulation number larger than 40. For such instances, one has to resort to heuristics and approximation algorithms. Here we present the algorithm CycleKiller which is the first approximation algorithm that can produce solutions verifiably close to optimality for instances with hundreds or even thousands of reticulations. Theoretically, the algorithm is an exponentialtime 2approximation (or 4approximation in its fastest mode). However, using simulations we demonstrate that in practice the algorithm runs quickly for large and difficult instances, producing solutions within one percent of optimality. An implementation of this algorithm, which extends the theoretical work of [14], has been made publicly available. © 2012 SpringerVerlag."








Katharina Huber,
Leo van Iersel,
Steven Kelk and
Radoslaw Suchecki. A Practical Algorithm for Reconstructing Level1 Phylogenetic Networks. In TCBB, Vol. 8(3):607620, 2011. Keywords: explicit network, from triplets, galled tree, generation, heuristic, phylogenetic network, phylogeny, Program LEV1ATHAN, Program Lev1Generator, reconstruction, software. Note: http://arxiv.org/abs/0910.4067.
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"Recently, much attention has been devoted to the construction of phylogenetic networks which generalize phylogenetic trees in order to accommodate complex evolutionary processes. Here, we present an efficient, practical algorithm for reconstructing level1 phylogenetic networksa type of network slightly more general than a phylogenetic treefrom triplets. Our algorithm has been made publicly available as the program Lev1athan. It combines ideas from several known theoretical algorithms for phylogenetic tree and network reconstruction with two novel subroutines. Namely, an exponentialtime exact and a greedy algorithm both of which are of independent theoretical interest. Most importantly, Lev1athan runs in polynomial time and always constructs a level1 network. If the data are consistent with a phylogenetic tree, then the algorithm constructs such a tree. Moreover, if the input triplet set is dense and, in addition, is fully consistent with some level1 network, it will find such a network. The potential of Lev1athan is explored by means of an extensive simulation study and a biological data set. One of our conclusions is that Lev1athan is able to construct networks consistent with a high percentage of input triplets, even when these input triplets are affected by a low to moderate level of noise. © 2011 IEEE."



Leo van Iersel and
Steven Kelk. Constructing the Simplest Possible Phylogenetic Network from Triplets. In ALG, Vol. 60(2):207235, 2011. Keywords: explicit network, from triplets, galled tree, level k phylogenetic network, minimum number, phylogenetic network, phylogeny, polynomial, Program Marlon, Program Simplistic. Note: http://dx.doi.org/10.1007/s0045300993330.
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"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. © 2009 The Author(s)."



Leo van Iersel and
Steven Kelk. When two trees go to war. In JTB, Vol. 269(1):245255, 2011. Keywords: APX hard, explicit network, from clusters, from rooted trees, from sequences, from triplets, level k phylogenetic network, minimum number, NP complete, phylogenetic network, phylogeny, polynomial, reconstruction. Note: http://arxiv.org/abs/1004.5332.
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"Rooted phylogenetic networks are used to model nontreelike evolutionary histories. Such networks are often constructed by combining trees, clusters, triplets or characters into a single network that in some welldefined sense simultaneously represents them all. We review these four models and investigate how they are related. Motivated by the parsimony principle, one often aims to construct a network that contains as few reticulations (nontreelike evolutionary events) as possible. In general, the model chosen influences the minimum number of reticulation events required. However, when one obtains the input data from two binary (i.e. fully resolved) trees, we show that the minimum number of reticulations is independent of the model. The number of reticulations necessary to represent the trees, triplets, clusters (in the softwired sense) and characters (with unrestricted multiple crossover recombination) are all equal. Furthermore, we show that these results also hold when not the number of reticulations but the level of the constructed network is minimised. We use these unification results to settle several computational complexity questions that have been open in the field for some time. We also give explicit examples to show that already for data obtained from three binary trees the models begin to diverge. © 2010 Elsevier Ltd."








Jaroslaw Byrka,
Pawel Gawrychowski,
Katharina Huber and
Steven Kelk. Worstcase optimal approximation algorithms for maximizing triplet consistency within phylogenetic networks. In Journal of Discrete Algorithms, Vol. 8(1):6575, 2010. Keywords: approximation, explicit network, from triplets, galled tree, level k phylogenetic network, phylogenetic network, phylogeny, reconstruction. Note: http://arxiv.org/abs/0710.3258.
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"The study of phylogenetic networks is of great interest to computational evolutionary biology and numerous different types of such structures are known. This article addresses the following question concerning rooted versions of phylogenetic networks. What is the maximum value of p ∈ [0, 1] such that for every input set T of rooted triplets, there exists some network N such that at least p  T  of the triplets are consistent with N? We call an algorithm that computes such a network (where p is maximum) worstcase optimal. Here we prove that the set containing all triplets (the full triplet set) in some sense defines p. Moreover, given a network N that obtains a fraction p′ for the full triplet set (for any p′), we show how to efficiently modify N to obtain a fraction ≥ p′ for any given triplet set T. We demonstrate the power of this insight by presenting a worstcase optimal result for level1 phylogenetic networks improving considerably upon the 5/12 fraction obtained recently by Jansson, Nguyen and Sung. For level2 phylogenetic networks we show that p ≥ 0.61. We emphasize that, because we are taking  T  as a (trivial) upper bound on the size of an optimal solution for each specific input T, the results in this article do not exclude the existence of approximation algorithms that achieve approximation ratio better than p. Finally, we note that all the results in this article also apply to weighted triplet sets. © 2009 Elsevier B.V. All rights reserved."



Leo van Iersel,
Steven Kelk,
Regula Rupp and
Daniel H. Huson. Phylogenetic Networks Do not Need to Be Complex: Using Fewer Reticulations to Represent Conflicting Clusters. In ISMB10, Vol. 26(12):i124i131 of BIO, 2010. Keywords: from clusters, level k phylogenetic network, Program Dendroscope, Program HybridInterleave, Program HybridNumber, reconstruction. Note: http://dx.doi.org/10.1093/bioinformatics/btq202, with proofs: http://arxiv.org/abs/0910.3082.
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"Phylogenetic trees are widely used to display estimates of how groups of species are evolved. Each phylogenetic tree can be seen as a collection of clusters, subgroups of the species that evolved from a common ancestor. When phylogenetic trees are obtained for several datasets (e.g. for different genes), then their clusters are often contradicting. Consequently, the set of all clusters of such a dataset cannot be combined into a single phylogenetic tree. Phylogenetic networks are a generalization of phylogenetic trees that can be used to display more complex evolutionary histories, including reticulate events, such as hybridizations, recombinations and horizontal gene transfers. Here, we present the new CASS algorithm that can combine any set of clusters into a phylogenetic network. We show that the networks constructed by CASS are usually simpler than networks constructed by other available methods. Moreover, we show that CASS is guaranteed to produce a network with at most two reticulations per biconnected component, whenever such a network exists. We have implemented CASS and integrated it into the freely available Dendroscope software. Contact: l.j.j.v.iersel@gmail.com. Supplementary information: Supplementary data are available at Bioinformatics online. © The Author(s) 2010. Published by Oxford University Press."






Leo van Iersel,
Steven Kelk and
Matthias Mnich. Uniqueness, intractability and exact algorithms: reflections on levelk phylogenetic networks. In JBCB, Vol. 7(4):597623, 2009. Keywords: explicit network, from triplets, galled tree, level k phylogenetic network, NP complete, phylogenetic network, phylogeny, reconstruction, uniqueness. Note: http://arxiv.org/pdf/0712.2932v2.










Leo van Iersel,
Judith Keijsper,
Steven Kelk,
Leen Stougie,
Ferry Hagen and
Teun Boekhout. Constructing level2 phylogenetic networks from triplets. In RECOMB08, Vol. 4955:450462 of LNCS, springer, 2008. Keywords: explicit network, from triplets, level k phylogenetic network, NP complete, phylogenetic network, phylogeny, polynomial, Program Level2, reconstruction. Note: http://homepages.cwi.nl/~iersel/level2full.pdf. An appendix with proofs can be found here http://arxiv.org/abs/0707.2890.
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"Jansson and Sung showed that, given a dense set of input triplets T (representing hypotheses about the local evolutionary relationships of triplets of taxa), it is possible to determine in polynomial time whether there exists a level1 network consistent with T, and if so, to construct such a network [24]. Here, we extend this work by showing that this problem is even polynomial time solvable for the construction of level2 networks. This shows that, assuming density, it is tractable to construct plausible evolutionary histories from input triplets even when such histories are heavily nontreelike. This further strengthens the case for the use of tripletbased methods in the construction of phylogenetic networks. We also implemented the algorithm and applied it to yeast data. © 2009 IEEE."



Leo van Iersel and
Steven Kelk. Constructing the Simplest Possible Phylogenetic Network from Triplets. In ISAAC08, Vol. 5369:472483 of LNCS, springer, 2008. Keywords: explicit network, from triplets, galled tree, level k phylogenetic network, minimum number, phylogenetic network, phylogeny, polynomial, Program Marlon, Program Simplistic. Note: http://arxiv.org/abs/0805.1859.



