Limits of Kernelization and Parametrization for Phylogenetic Diversity with Dependencies

TL;DR

This paper investigates the computational complexity of a conservation problem that combines phylogenetic diversity with predator-prey dependencies, revealing hardness results and kernelization limits in parameterized complexity.

Contribution

It introduces the $ ext{α}$-PDD problem, analyzes its parameterized complexity, and establishes new hardness and kernelization bounds, including fixed-parameter tractability and non-existence of polynomial kernels.

Findings

$ ext{α}$-PDD is W[1]-hard when parameterized by solution size or diversity threshold.

$ ext{α}$-PDD is fixed-parameter tractable when parameterized by vertex cover number.

No polynomial kernel exists for $ ext{α}$-PDD when parameterized by vertex cover plus diversity threshold.

Abstract

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 ($\epsilon$-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 $\alpha$-PDD problem alters PDD by requiring that at least some fraction $\alpha$ of the prey of every saved species are also saved. In this paper, we study the parameterized complexity of $\alpha$-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, $\alpha$-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 $\alpha$-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 $\alpha$-PDD also when parameterized by a range of structural parameters of the food web, such as its dist[...]