Limits of Kernelization and Parametrization for Phylogenetic Diversity with Dependencies

TL;DR

This paper proves 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, and proves that, when parameterized by the distance to clique, 1-PDD admits a linear kernel.

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 ($ε$-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[...]

Figures (2)

• Figure 1: The relationship between structural parameters of the food web and the complexity of solving $1$-PDD (top left) and $1$-PDD$_{\text{s}}$ (bottom right). A parameter $p$ is marked red () if the problem is NP-hard for constant values of $p$. If the problem is FPT with respect to $p$, then $p$ is marked in orange (), green () or blue (). Green means that the problem has a polynomial kernel, orange means that it does not, and blue means that kernelization remains open. An edge $p_1 p_2$ between parameters $p_1$ and $p_2$ indicates that, in every graph, the parameter $p_1$ can be bounded by a function of $p_2$. A more in-depth look into the hierarchy of graph parameters can be found in graphparameters.
• Figure 2: The cross-composition from $t \in \{9,\ldots,16\}$ clique instances to 1-PDD. As illustrated for one vertex $\{u, v\} \in E_1$, every vertex $\{x, y\} \in E_i$, for $i \in [t-1]_0$, has in-edges from the endpoints $x, y \in V$ of the corresponding edge, and from those vertices among $0_0, 1_0, \ldots, 0_{\log t}, 1_{\log t}$ that encode the index $i$.

Theorems & Definitions (7)

• Theorem 1: cygan
• Lemma 4
• Lemma 5
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10