核化与参数化在存在依赖关系的系统多样性中的极限
Limits of Kernelization and Parametrization for Phylogenetic Diversity with Dependencies
AI总结：
本文研究了在存在依赖关系的系统多样性优化问题中，参数化复杂性及核化限制，证明了α-PDD在某些参数下不具有多项式核。
摘要：
在最大化系统多样性问题中，我们被给定一个表示物种遗传亲缘关系的系统树，并被要求选择一个最大系统多样性物种子集以通过保护努力得以保存，但受到预算限制，只能保存k个物种。这忽略了如果未保存其捕食的物种，则保存捕食者物种是无用的。因此，在优化系统多样性与依赖关系（ε-PDD）问题中，我们还被给出一个表示物种捕食者-被捕食者关系的生态网络。目标是保存k个物种，使得对于每个被保存的物种，至少有一个其被捕食者也被保存。该问题即使当系统树是星形时仍然是NP难的。α-PDD问题通过要求至少保存每个被保存物种的某些比例α的被捕食者来改变PDD。在本文中，我们研究了α-PDD的参数化复杂性。我们证明当参数化为解决方案大小k、多样性阈值D或其补集时，该问题属于W[1]-难和XP。当参数化为生态网络的顶点覆盖数时，α-PDD是固定参数可解的（FPT）。FPT问题计算难度的一个关键度量是可获得的最小核的大小。我们证明，当参数化为距离到clique时，1-PDD允许线性核。我们的主要贡献是证明当系统树是星形时，即使参数化为顶点覆盖数加上多样性阈值D，α-PDD也不允许多项式核。这表明当参数化为生态网络的结构参数，如其dist时，α-PDD也不允许多项式核。
英文摘要：
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[...]
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