Maximizing Phylogenetic Diversity under Ecological Constraints: A Parameterized Complexity Study
<|reference_start|>Maximizing Phylogenetic Diversity under Ecological Constraints: A Parameterized Complexity Study: 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.<|reference_end|>
@article{komusiewicz2024maximizing,
    title={Maximizing Phylogenetic Diversity under Ecological Constraints: A
  Parameterized Complexity Study},
    author={Christian Komusiewicz and Jannik Schestag},
    journal={arXiv preprint arXiv:2405.17314},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.17314},
    primaryClass={cs.CC}
}