A common problem in phylogenetics is to try to infer a species phylogeny from gene trees. We consider different variants of this problem. The first variant, called Unrestricted Minimal Episodes Inference, aims at inferring a species tree based on a model with speciation and duplication where duplications are clustered in duplication episodes. The goal is to minimize the number of such episodes. The second variant, Parental Hybridization, aims at inferring a species network based on a model with speciation and reticulation. The goal is to minimize the number of reticulation events. It is a variant of the well-studied Hybridization Number problem with a more generous view on which gene trees are consistent with a given species network. We show that these seemingly different problems are in fact closely related and can, surprisingly, both be solved in polynomial time, using a structure we call "beaded trees". However, we also show that methods based on these problems have to be used with care because the optimal species phylogenies always have a restricted form. To mitigate this problem, we introduce a new variant of Unrestricted Minimal Episodes Inference that minimizes the duplication episode depth. We prove that this new variant of the problem can also be solved in polynomial time.

中文翻译：

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涉及重复和网状结构的系统发生推理问题的多项式时间算法。

系统发育学中的一个普遍问题是试图从基因树中推断出一个物种系统发育。我们考虑此问题的不同变体。第一个变种称为无限制最小情节推断，旨在基于具有物种形成和重复的模型来推断物种树，其中重复项聚集在重复情节中。目标是最大程度地减少此类事件的发生。第二种变体，父母亲杂交，旨在基于具有物种形成和网状形成的模型来推断物种网络。目标是最大程度地减少网状事件的数量。它是经过充分研究的“杂交数”问题的一种变体，对于哪些基因树与给定物种网络一致，它具有更大的看法。我们表明，这些看似不同的问题实际上密切相关，并且令人惊讶的是，都可以在多项式时间内使用我们称为“串珠树”的结构来求解。但是，我们还表明，基于这些问题的方法必须谨慎使用，因为最佳物种系统发育始终具有受限的形式。为了缓解此问题，我们引入了不受限制的最小情节推断的新变体，该变体可以最大程度地减少重复情节的深度。我们证明问题的这个新变体也可以在多项式时间内解决。