AI 中文总结
研究最大全路径系统发育多样性（MapPD）问题，证明其在以解中包含或排除的物种数量为参数时分别是W[2]难和W[1]难，同时表明对多样性阈值等是固定参数可处理的，还给出相关算法及多项式核化。
AI 中文摘要
系统发育多样性（PD）是生物多样性的基本度量，最初定义于系统发育树，在保护生物学中广泛应用。系统发育树常被推广为有向无环图即系统发育网络，因此需要对PD进行相应推广。边加权系统发育网络的自然推广是全路径度量，其中一组物种（分类单元）S的多样性定义为从根到S中至少一个物种的路径上所有边的总权重。虽然在树上最大化PD可在多项式时间内解决，但网络上的相应问题是NP难且难以近似。我们对最大全路径PD（MapPD）问题进行了系统的参数化复杂性研究。我们确定了以解中包含的物种数量为参数时的W[2]难，以及以排除的物种数量为参数时的W[1]难。积极的一面是，我们表明该问题对于多样性阈值和可接受的多样性损失是固定参数可处理的。我们进一步分析了网络与树的接近程度如何影响算法行为，并给出了以网状化数量和基础图树宽为参数的单指数固定参数算法。最后，我们给出了关于网状化边数量的MapPD的多项式核化。
英文摘要
Phylogenetic Diversity (PD) is a fundamental measure of biodiversity, originally defined on phylogenetic trees and widely used in conservation biology. Phylogenetic trees are often generalised to directed acyclic graphs, called phylogenetic networks. As such, a corresponding generalization of PD is needed. A natural generalization to edge-weighted phylogenetic networks is the all-paths measure, where the diversity of a set S of species (taxa) is defined as the total weight of all edges that lie on a path from the root to at least one species in S. While maximizing PD on trees can be solved in polynomial time, the corresponding problem on networks is NP-hard and difficult to approximate. We undertake a systematic parameterized complexity study of the Max-All-Paths-PD (MapPD) problem. We establish W[2]-hardness when parameterized by the number of species that are included in a solution, and W[1]-hardness for the number of species that are excluded. On the positive side, we show that the problem is fixed-parameter tractable with respect to the threshold of diversity and the acceptable loss of diversity. We further analyze how the network's proximity to a tree influences algorithmic behavior and present single-exponential fixed-parameter algorithms when parameterized by the number of reticulations and by the treewidth of the underlying graph. Finally, we present a polynomial kernelization for MapPD with respect to the number of reticulation edges.
CommentsA previous version was presented at IPEC 2023; now submitted to a journal
