Maximizing Phylogenetic Diversity under Ecological Constraints: A Parameterized Complexity Study
Christian Komusiewicz, Jannik Schestag
TL;DR
This work investigates the computational complexity of Optimizing PD with Dependencies (PDD) and its star variant s-PDD under parameterized complexity. The authors establish that PDD is fixed-parameter tractable (FPT) when parameterized by the combination of the solution size $k$ and the phylogenetic tree height or by the diversity threshold $D$, and they show W[1]-hardness with respect to the diversity-loss parameter ${\overline{D}}$. They also analyze structural parameters of the food-web, proving that s-PDD is FPT when parameterized by the distance to a co-cluster/cluster graph and that s-PDD is FPT with respect to the treewidth of the food-web, thereby refuting a conjecture that PDD remains NP-hard on trees. The paper employs color-coding, pattern-based reductions, and dynamic programming on tree decompositions to achieve these results, providing both algorithmic techniques and complexity boundaries. Overall, the findings delineate natural, ecologically motivated parameters under which viable, diversity-maximizing conservation strategies can be computed efficiently and identify key open questions for further study.
Abstract
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.