In the Maximize Phylogenetic Diversity problem, we are given a phylogenetic tree that represents the genetic proximity of species, and we are asked to select a subset of species of maximum phylogenetic diversity to be preserved through conservation efforts, subject to budgetary constraints that allow only k species to be saved. This neglects that it is futile to preserve a predatory species if we do not also preserve at least a subset of the prey it feeds on. Thus, in the Optimizing PD with Dependencies ($ε$-PDD) problem, we are additionally given a food web that represents the predator-prey relationships between species. The goal is to save a set of k species of maximum phylogenetic diversity such that for every saved species, at least one of its prey is also saved. This problem is NP-hard even when the phylogenetic tree is a star. The $α$-PDD problem alters PDD by requiring that at least some fraction $α$ of the prey of every saved species are also saved. In this paper, we study the parameterized complexity of $α$-PDD. We prove that the problem is W[1]-hard and in XP when parameterized by the solution size k, the diversity threshold D, or their complements. When parameterized by the vertex cover number of the food web, $α$-PDD is fixed-parameter tractable (FPT). A key measure of the computational difficulty of a problem that is FPT is the size of the smallest kernel that can be obtained. We prove that, when parameterized by the distance to clique, 1-PDD admits a linear kernel. Our main contribution is to prove that $α$-PDD does not admit a polynomial kernel when parameterized by the vertex cover number plus the diversity threshold D, even if the phylogenetic tree is a star. This implies the non-existence of a polynomial kernel for $α$-PDD also when parameterized by a range of structural parameters of the food web, such as its dist[...]

翻译：在最大化系统发育多样性问题中，我们获得一棵表示物种间遗传亲缘关系的系统发育树，并需要选择具有最大系统发育多样性的物种子集以通过保护措施进行保存，同时受限于预算约束——仅允许保存k个物种。这忽略了以下事实：若不同时保存捕食者所依赖的至少部分猎物物种，则单独保存该捕食者物种是徒劳的。因此，在具有依赖关系的优化系统发育多样性（$ε$-PDD）问题中，我们额外获得一个表示物种间捕食者-猎物关系的食物网。目标在于保存一组具有最大系统发育多样性的k个物种，且要求每个被保存物种至少有一个猎物物种也被保存。即使系统发育树为星形结构，该问题仍是NP难的。$α$-PDD问题通过要求每个被保存物种需保存其至少$α$比例的猎物物种来修改PDD问题。本文研究$α$-PDD问题的参数化复杂度。我们证明当以解规模k、多样性阈值D或其补集作为参数时，该问题是W[1]难的且属于XP类。当以食物网的顶点覆盖数作为参数时，$α$-PDD是固定参数可处理（FPT）的。衡量FPT问题计算难度的关键指标是可获得的最小核的规模。我们证明当以到团的距离作为参数时，1-PDD存在线性核。我们的主要贡献在于证明：即使系统发育树为星形结构，当以顶点覆盖数与多样性阈值D之和作为参数时，$α$-PDD不存在多项式核。这进一步意味着当以食物网的一系列结构参数（如其距离...）作为参数时，$α$-PDD同样不存在多项式核。