
Magnus Bordewich,
Simone Linz,
Katherine St. John and
Charles Semple. A reduction algorithm for computing the hybridization number of two trees. In EBIO, Vol. 3:8698, 2007. Keywords: agreement forest, FPT, from rooted trees, hybridization, phylogenetic network, phylogeny, Program HybridNumber. Note: http://www.math.canterbury.ac.nz/~c.semple/papers/BLSS07.pdf.





Charles Choy,
Jesper Jansson,
Kunihiko Sadakane and
WingKin Sung. Computing the maximum agreement of phylogenetic networks. In TCS, Vol. 335(1):93107, 2005. Keywords: dynamic programming, FPT, level k phylogenetic network, MASN, NP complete, phylogenetic network, phylogeny. Note: http://www.df.lth.se/~jj/Publications/masn8_TCS2005.pdf.
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"We introduce the maximum agreement phylogenetic subnetwork problem (MASN) for finding branching structure shared by a set of phylogenetic networks. We prove that the problem is NPhard even if restricted to three phylogenetic networks and give an O(n2)time algorithm for the special case of two level1 phylogenetic networks, where n is the number of leaves in the input networks and where N is called a levelf phylogenetic network if every biconnected component in the underlying undirected graph induces a subgraph of N containing at most f nodes with indegree 2. We also show how to extend our technique to yield a polynomialtime algorithm for any two levelf phylogenetic networks N1,N2 satisfying f=O(logn); more precisely, its running time is O(V(N1)·V(N2)·2f1+f2), where V(Ni) and fi denote the set of nodes in Ni and the level of Ni, respectively, for i∈{1,2}. © 2005 Elsevier B.V. All rights reserved."





Josh Voorkamp né Collins,
Simone Linz and
Charles Semple. Quantifying hybridization in realistic time. In JCB, Vol. 18(10):13051318, 2011. Keywords: explicit network, FPT, from rooted trees, hybridization, minimum number, phylogenetic network, phylogeny, Program HybridInterleave, reconstruction, software. Note: http://wwwcsif.cs.ucdavis.edu/~linzs/CLS10_interleave.pdf, software available at http://www.math.canterbury.ac.nz/~c.semple/software.shtml.
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"Recently, numerous practical and theoretical studies in evolutionary biology aim at calculating the extent to which reticulationfor example, horizontal gene transfer, hybridization, or recombinationhas influenced the evolution for a set of presentday species. It has been shown that inferring the minimum number of hybridization events that is needed to simultaneously explain the evolutionary history for a set of trees is an NPhard and also fixedparameter tractable problem. In this article, we give a new fixedparameter algorithm for computing the minimum number of hybridization events for when two rooted binary phylogenetic trees are given. This newly developed algorithm is based on interleavinga technique using repeated kernelization steps that are applied throughout the exhaustive search part of a fixedparameter algorithm. To show that our algorithm runs efficiently to be applicable to a wide range of practical problem instances, we apply it to a grass data set and highlight the significant improvements in terms of running times in comparison to an algorithm that has previously been implemented. © 2011, Mary Ann Liebert, Inc."



Ali Tofigh,
Mike Hallett and
Jens Lagergren. Simultaneous Identification of Duplications and Lateral Gene Transfers. In TCBB, Vol. 8(2):517535, 2011. Keywords: duplication, explicit network, FPT, from rooted trees, from species tree, lateral gene transfer, loss, NP complete, phylogenetic network, phylogeny, reconstruction. Note: http://dx.doi.org/10.1109/TCBB.2010.14.
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"The incongruency between a gene tree and a corresponding species tree can be attributed to evolutionary events such as gene duplication and gene loss. This paper describes a combinatorial model where socalled DTLscenarios are used to explain the differences between a gene tree and a corresponding species tree taking into account gene duplications, gene losses, and lateral gene transfers (also known as horizontal gene transfers). The reasonable biological constraint that a lateral gene transfer may only occur between contemporary species leads to the notion of acyclic DTLscenarios. Parsimony methods are introduced by defining appropriate optimization problems. We show that finding most parsimonious acyclic DTLscenarios is NPhard. However, by dropping the condition of acyclicity, the problem becomes tractable, and we provide a dynamic programming algorithm as well as a fixedparameter tractable algorithm for finding most parsimonious DTLscenarios. © 2011 IEEE."



ZhiZhong Chen and
Lusheng Wang. HybridNET: a tool for constructing hybridization networks. In BIO, Vol. 26(22):29122913, 2010. Keywords: agreement forest, FPT, from rooted trees, hybridization, phylogenetic network, phylogeny, Program HybridNET, software. Note: http://rnc.r.dendai.ac.jp/~chen/papers/note2.pdf.
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"Motivations: When reticulation events occur, the evolutionary history of a set of existing species can be represented by a hybridization network instead of an evolutionary tree. When studying the evolutionary history of a set of existing species, one can obtain a phylogenetic tree of the set of species with high confidence by looking at a segment of sequences or a set of genes. When looking at another segment of sequences, a different phylogenetic tree can be obtained with high confidence too. This indicates that reticulation events may occur. Thus, we have the following problem: given two rooted phylogenetic trees on a set of species that correctly represent the treelike evolution of different parts of their genomes, what is the hybridization network with the smallest number of reticulation events to explain the evolution of the set of species under consideration? Results: We develop a program, named HybridNet, for constructing a hybridization network with the minimum number of reticulate vertices from two input trees. We first implement the O(3dn)time algorithm by Whidden et al. for computing a maximum (acyclic) agreement forest. Our program can output all the maximum (acyclic) agreement forests. We then augment the program so that it can construct an optimal hybridization network for each given maximum acyclic agreement forest. To our knowledge, this is the first time that optimal hybridization networks can be rapidly constructed. © The Author 2010. Published by Oxford University Press. All rights reserved."



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."



Celine Scornavacca,
Simone Linz and
Benjamin Albrecht. A first step towards computing all hybridization networks for two rooted binary phylogenetic trees. In JCB, Vol. 19:12271242, 2012. Keywords: agreement forest, explicit network, FPT, from rooted trees, phylogenetic network, phylogeny, Program Dendroscope, Program Hybroscale, reconstruction. Note: http://arxiv.org/abs/1109.3268.
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"Recently, considerable effort has been put into developing fast algorithms to reconstruct a rooted phylogenetic network that explains two rooted phylogenetic trees and has a minimum number of hybridization vertices. With the standard app1235roach to tackle this problem being combinatorial, the reconstructed network is rarely unique. From a biological point of view, it is therefore of importance to not only compute one network, but all possible networks. In this article, we make a first step toward approaching this goal by presenting the first algorithmcalled allMAAFsthat calculates all maximumacyclicagreement forests for two rooted binary phylogenetic trees on the same set of taxa. © Copyright 2012, Mary Ann Liebert, Inc. 2012."



Leo van Iersel and
Simone Linz. A quadratic kernel for computing the hybridization number of multiple trees. In IPL, Vol. 113:318323, 2013. Keywords: explicit network, FPT, from rooted trees, kernelization, minimum number, phylogenetic network, phylogeny, Program Clustistic, Program MaafB, Program PIRN, reconstruction. Note: http://arxiv.org/abs/1203.4067, poster.
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"It has recently been shown that the NPhard problem of calculating the minimum number of hybridization events that is needed to explain a set of rooted binary phylogenetic trees by means of a hybridization network is fixedparameter tractable if an instance of the problem consists of precisely two such trees. In this paper, we show that this problem remains fixedparameter tractable for an arbitrarily large set of rooted binary phylogenetic trees. In particular, we present a quadratic kernel. © 2013 Elsevier B.V."



Chris Whidden,
Robert G. Beiko and
Norbert Zeh. FixedParameter Algorithms for Maximum Agreement Forests. In SICOMP, Vol. 42(4):14311466, 2013. Keywords: agreement forest, explicit network, FPT, from rooted trees, hybridization, minimum number, phylogenetic network, phylogeny, Program HybridInterleave, reconstruction, SPR distance. Note: http://arxiv.org/abs/1108.2664, slides.
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"We present new and improved fixedparameter algorithms for computing maximum agreement forests of pairs of rooted binary phylogenetic trees. The size of such a forest for two trees corresponds to their subtree pruneandregraft distance and, if the agreement forest is acyclic, to their hybridization number. These distance measures are essential tools for understanding reticulate evolution. Our algorithm for computing maximum acyclic agreement forests is the first depthbounded search algorithm for this problem. Our algorithms substantially outperform the best previous algorithms for these problems. © 2013 Society for Industrial and Applied Mathematics."



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."



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.



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.



ZhiZhong Chen,
Fei Deng and
Lusheng Wang. Simultaneous Identification of Duplications, Losses, and Lateral Gene Transfers. In TCBB, Vol. 9(5):15151528, 2012. Keywords: duplication, explicit network, FPT, from rooted trees, from species tree, lateral gene transfer, loss, phylogenetic network, phylogeny, reconstruction. Note: http://www.cs.cityu.edu.hk/~lwang/research/tcbb2012c.pdf.
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"We give a fixedparameter algorithm for the problem of enumerating all minimumcost LCAreconciliations involving gene duplications, gene losses, and lateral gene transfers (LGTs) for a given species tree S and a given gene tree G. Our algorithm can work for the weighted version of the problem, where the costs of a gene duplication, a gene loss, and an LGT are left to the user's discretion. The algorithm runs in O(m+3 k/c n) time, where m is the number of vertices in S, n is the number of vertices in G, c is the smaller between a gene duplication cost and an LGT cost, and k is the minimum cost of an LCAreconciliation between S and G. The time complexity is indeed better if the cost of a gene loss is greater than 0. In particular, when the cost of a gene loss is at least 0.614c, the running time of the algorithm is O(m+2.78 k/cn). © 20042012 IEEE."



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.



Katharina Huber,
Leo van Iersel,
Vincent Moulton,
Celine Scornavacca and
Taoyang Wu. Reconstructing phylogenetic level1 networks from nondense binet and trinet sets. In ALG, Vol. 77(1):173200, 2017. Keywords: explicit network, FPT, from binets, from trinets, NP complete, phylogenetic network, phylogeny, polynomial, reconstruction. Note: http://arxiv.org/abs/1411.6804.



Benjamin Albrecht. Computing all hybridization networks for multiple binary phylogenetic input trees. In BMCB, Vol. 16(236):115, 2015. Keywords: agreement forest, explicit network, exponential algorithm, FPT, from rooted trees, phylogenetic network, phylogeny, Program Hybroscale, Program PIRN, reconstruction. Note: http://dx.doi.org/10.1186/s1285901506607.



Maria Anaya,
Olga AnipchenkoUlaj,
Aisha Ashfaq,
Joyce Chiu,
Mahedi Kaiser,
Max Shoji Ohsawa,
Megan Owen,
Ella Pavlechko,
Katherine St. John,
Shivam Suleria,
Keith Thompson and
Corrine Yap. On Determining if Treebased Networks Contain Fixed Trees. In BMB, Vol. 78(5):961969, 2016. Keywords: explicit network, FPT, NP complete, phylogenetic network, phylogeny, treebased network. Note: http://arxiv.org/abs/1602.02739.



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.





Charles Choy,
Jesper Jansson,
Kunihiko Sadakane and
WingKin Sung. Computing the maximum agreement of phylogenetic networks. In Proceedings of Computing: the Tenth Australasian Theory Symposium (CATS'04), Vol. 91:134147 of Electronic Notes in Theoretical Computer Science, 2004. Keywords: dynamic programming, FPT, level k phylogenetic network, MASN, NP complete, phylogenetic network, phylogeny. Note: http://www.df.lth.se/~jj/Publications/masn6_CATS2004.pdf.
Toggle abstract
"We introduce the maximum agreement phylogenetic subnetwork problem (MASN) of finding a branching structure shared by a set of phylogenetic networks. We prove that the problem is NPhard even if restricted to three phylogenetic networks and give an O(n2)time algorithm for the special case of two level1 phylogenetic networks, where n is the number of leaves in the input networks and where N is called a levelf phylogenetic network if every biconnected component in the underlying undirected graph contains at most f nodes having indegree 2 in N. Our algorithm can be extended to yield a polynomialtime algorithm for two levelf phylogenetic networks N 1,N2 for any f which is upperbounded by a constant; more precisely, its running time is O(V(N1)·V(N 2)·4f), where V(Ni) denotes the set of nodes of Ni. © 2004 Published by Elsevier B.V."



Daniel H. Huson and
Tobias Kloepper. Beyond Galled Trees  Decomposition and Computation of Galled Networks. In RECOMB07, Vol. 4453:211225 of LNCS, springer, 2007. Keywords: FPT, from splits, from trees, galled network, phylogenetic network, phylogeny, Program SplitsTree, reconstruction. Note: http://dx.doi.org/10.1007/9783540716815_15, errata..



Mike Hallett,
Jens Lagergren and
Ali Tofigh. Simultaneous Identification of Duplications and Lateral Transfers. In RECOMB04, Pages 347356, 2004. Keywords: duplication, explicit network, FPT, from rooted trees, from species tree, lateral gene transfer, loss, NP complete, parsimony, phylogenetic network, phylogeny, polynomial, reconstruction. Note: http://www.nada.kth.se/~jensl/p164hallett.pdf.



Daniel H. Huson,
Regula Rupp,
Vincent Berry,
Philippe Gambette and
Christophe Paul. Computing Galled Networks from Real Data. In ISMBECCB09, Vol. 25(12):i85i93 of BIO, 2009. Keywords: abstract network, cluster containment, explicit network, FPT, from clusters, from rooted trees, galled network, NP complete, phylogenetic network, phylogeny, polynomial, Program Dendroscope, reconstruction. Note: http://hallirmm.ccsd.cnrs.fr/lirmm00368545/en/.
Toggle abstract
"Motivation: Developing methods for computing phylogenetic networks from biological data is an important problem posed by molecular evolution and much work is currently being undertaken in this area. Although promising approaches exist, there are no tools available that biologists could easily and routinely use to compute rooted phylogenetic networks on real datasets containing tens or hundreds of taxa. Biologists are interested in clades, i.e. groups of monophyletic taxa, and these are usually represented by clusters in a rooted phylogenetic tree. The problem of computing an optimal rooted phylogenetic network from a set of clusters, is hard, in general. Indeed, even the problem of just determining whether a given network contains a given cluster is hard. Hence, some researchers have focused on topologically restricted classes of networks, such as galled trees and levelk networks, that are more tractable, but have the practical drawback that a given set of clusters will usually not possess such a representation. Results: In this article, we argue that galled networks (a generalization of galled trees) provide a good tradeoff between level of generality and tractability. Any set of clusters can be represented by some galled network and the question whether a cluster is contained in such a network is easy to solve. Although the computation of an optimal galled network involves successively solving instances of two different NPcomplete problems, in practice our algorithm solves this problem exactly on large datasets containing hundreds of taxa and many reticulations in seconds, as illustrated by a dataset containing 279 prokaryotes. © 2009 The Author(s)."



Chris Whidden,
Robert G. Beiko and
Norbert Zeh. Fast FPT Algorithms for Computing Rooted Agreement Forests: Theory and Experiments. In Proceedings of the ninth International Symposium on Experimental Algorithms (SEA'10), Vol. 6049:141153 of LNCS, springer, 2010. Keywords: agreement forest, explicit network, FPT, from rooted trees, hybridization, minimum number, phylogenetic network, phylogeny, Program HybridInterleave, reconstruction, SPR distance. Note: https://www.cs.dal.ca/sites/default/files/technical_reports/CS201003.pdf.
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"We improve on earlier FPT algorithms for computing a rooted maximum agreement forest (MAF) or a maximum acyclic agreement forest (MAAF) of a pair of phylogenetic trees. Their sizes give the subtreepruneandregraft (SPR) distance and the hybridization number of the trees, respectively. We introduce new branching rules that reduce the running time of the algorithms from O(3 kn) and O(3 kn log n) to O(2.42 kn) and O(2.42 kn log n), respectively. In practice, the speed up may be much more than predicted by the worstcase analysis.We confirm this intuition experimentally by computing MAFs for simulated trees and trees inferred from protein sequence data. We show that our algorithm is orders of magnitude faster and can handle much larger trees and SPR distances than the best previous methods, treeSAT and sprdist. © SpringerVerlag Berlin Heidelberg 2010."





Chris Whidden and
Norbert Zeh. A Unifying View on Approximation and FPT of Agreement Forests. In WABI09, Vol. 5724:390402 of LNCS, Springer, 2009. Keywords: agreement forest, approximation, explicit network, FPT, minimum number, phylogenetic network, phylogeny, reconstruction. Note: https://www.cs.dal.ca/sites/default/files/technical_reports/CS200902.pdf.
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"We provide a unifying view on the structure of maximum (acyclic) agreement forests of rooted and unrooted phylogenies. This enables us to obtain linear or O(n log n)time 3approximation and improved fixedparameter algorithms for the subtree prune and regraft distance between two rooted phylogenies, the tree bisection and reconnection distance between two unrooted phylogenies, and the hybridization number of two rooted phylogenies. © 2009 Springer Berlin Heidelberg."



