In the NP-hard Optimizing PD with Dependencies (PDD) problem, the input consists of a phylogenetic tree $T$ over a set of taxa $X$, a food-web that describes the prey-predator relationships in $X$, and integers $k$ and $D$. The task is to find a set $S$ of $k$ species that is viable in the food-web such that the subtree of $T$ obtained by retaining only the vertices of $S$ has total edge weight at least $D$. Herein, viable means that for every predator taxon of $S$, the set $S$ contains at least one prey taxon. We provide the first systematic analysis of PDD and its special case s-PDD from a parameterized complexity perspective. For solution-size related parameters, we show that PDD is FPT with respect to $D$ and with respect to $k$ plus the height of the phylogenetic tree. Moreover, we consider structural parameterizations of the food-web. For example, we show an FPT-algorithm for the parameter that measures the vertex deletion distance to graphs where every connected component is a complete graph. Finally, we show that s-PDD admits an FPT-algorithm for the treewidth of the food-web. This disproves a conjecture of Faller et al. [Annals of Combinatorics, 2011] who conjectured that s-PDD is NP-hard even when the food-web is a tree.

翻译：在NP难问题“带依赖关系的系统发育多样性优化”（PDD）中，输入包含一个在分类单元集$X$上的系统发育树$T$、描述$X$中捕食-被捕食关系的食物网，以及整数$k$和$D$。任务是寻找一个在食物网中可行的$k$个物种集合$S$，使得仅保留$S$中顶点所获得的$T$的子树的边权重总和至少为$D$。此处“可行”指对于$S$中的每个捕食者分类单元，集合$S$至少包含一个被捕食者分类单元。我们首次从参数化复杂性角度对PDD及其特例s-PDD进行了系统分析。对于解大小相关参数，我们证明PDD关于$D$以及关于$k$加系统发育树高度是固定参数可解（FPT）的。此外，我们考虑了食物网的结构参数化。例如，我们针对度量到每个连通分量均为完全图的图的顶点删除距离这一参数给出了FPT算法。最后，我们证明s-PDD对于食物网的树宽存在FPT算法。这推翻了Faller等人[Annals of Combinatorics, 2011]的猜想，他们曾推测即使食物网是树结构时s-PDD仍是NP难问题。