In [2] it was proven that the Cass algorithm is a polynomial-time algorithm for constructing level<=2 networks from clusters. Here we demonstrate, for each k>=0, a polynomial-time algorithm for constructing level-k phylogenetic networks from clusters. Unlike Cass the algorithm scheme given here is only of theoretical interest. It does, however, strengthen the hope that efficient polynomial-time algorithms (and perhaps fixed parameter tractable algorithms) exist for this problem.
Constructing minimal phylogenetic networks from softwired clusters is fixed parameter tractable
在这里，我们表明，给定一组分类群X上的一组聚类C，其中|X| =n，可以在时间f（k）.poly（n）中确定是否存在一个在软件意义上代表C中所有聚类的level <= k网络（即每个双连通分量的网状数至多为k的网络），如果是这样，则可以构造这样的网络。这扩展了Kelk，Scornavaca and Van Iersel（2011）的“On the elusiveness of clusters”的多项式时间结果。通过将“level-k generator”的概念推广到一般网络，我们将这个固定参数的易处理性结果推广到k不是指网络的水平而是指整个网络的网状数的问题。
Evolutionary histories for species that cross with one another or exchange genetic material can be represented by leaf-labelled, directed graphs called phylogenetic networks. A major challenge in the burgeoning area of phylogenetic networks is to develop algorithms for building such networks by amalgamating small networks into a single large network. The level of a phylogenetic network is a measure of its deviation from being a tree; the higher the level of network, the less treelike it becomes. Various algorithms have been developed for building level-1 networks from small networks. However, …
Constructing Phylogenetic Networks via Cherry Picking and Machine Learning
University of Trieste（的里雅斯特大学）；Delft Institute of Applied Mathematics（代尔夫特应用数学研究所）；CWI（荷兰国家数学与计算机科学研究中心）；Vrije Universiteit（自由大学）；INRIA-Erable（法国国家信息与自动化研究所Erable团队）
Here we show that deciding whether two rooted binary phylogenetic trees on the same set of taxa permit a cherry-picking sequence, a special type of elimination order on the taxa, is NP-complete. This improves on an earlier result which proved hardness for eight or more trees. Via a known equivalence between cherry-picking sequences and temporal phylogenetic networks, our result proves that it is NP-complete to determine the existence of a temporal phylogenetic network that contains topological embeddings of both trees. The hardness result also greatly strengthens previous inapproximability res…
A rooted phylogenetic network is a directed acyclic graph with a single root, whose sinks correspond to a set of species. As such networks are useful for representing the evolution of species that have undergone reticulate evolution, there has been great interest in developing the theory behind and algorithms for constructing them. However, unlike evolutionary trees, these networks can be highly non-planar, which can make them difficult to visualise and interpret. Here we investigate properties of planar rooted phylogenetic networks and algorithms for deciding whether or not rooted networks ha…
We present the first fixed-parameter algorithm for constructing a tree-child phylogenetic network that displays an arbitrary number of binary input trees and has the minimum number of reticulations among all such networks. The algorithm uses the recently introduced framework of cherry picking sequences and runs in $O((8k)^k \mathrm{poly}(n, t))$ time, where $n$ is the number of leaves of every tree, $t$ is the number of trees, and $k$ is the reticulation number of the constructed network. Moreover, we provide an efficient parallel implementation of the algorithm and show that it can deal with …
Background. The supertree problem, i.e., the task of finding a common refinement of a set of rooted trees is an important topic in mathematical phylogenetics. The special case of a common leaf set $L$ is known to be solvable in linear time. Existing approaches refine one input tree using information of the others and then test whether the results are isomorphic. Results. A linear-time algorithm, LinCR, for constructing the common refinement $T$ of $k$ input trees with a common leaf set is proposed that explicitly computes the parent function of $T$ in a bottom-up approach. Conclusion. LinCR is…
Phylogenetic trees are often constructed by using a metric on the set of taxa that label the leaves of the tree. While there are a number of methods for constructing a tree using a given metric, such trees will only display the metric if it satisfies the so-called "four point condition", established by Buneman in 1971. While this condition guarantees that a unique tree will display the metric, meaning that the distance between any two leaves can be found by adding the distances on arcs in the path between the leaves, it doesn't exclude the possibility that a phylogenetic network might also dis…
