Weighted Food Webs Make Computing Phylogenetic Diversity So Much Harder

TL;DR

This work introduces Weighted-PDD, a weighted generalization of phylogenetic diversity optimization under predator–prey dependencies, and proves its NP-hardness even on very simple structures. To cope with intractability, it defines rw-PDD (restricted prey requirements) and analyzes the problem through parameterized complexity with structural graph parameters of the food web. The authors establish XP and FPT results for rw-PDD and 1-PDD variants with respect to vertex cover, distance to cluster, and treewidth, employing a mix of dynamic programming on phylogenetic trees and reductions (including a Knapsack-based hardness proof). The results illuminate which ecological-network structures permit efficient optimization and provide practical pathways for conservation planning under realistic weighted interactions. Overall, the work advances both the theoretical understanding and algorithmic toolkit for maximizing phylogenetic diversity in ecologically constrained settings, while outlining open questions for more complex interaction models.

Abstract

Phylogenetic trees represent certain species and their likely ancestors. In such a tree, present-day species are leaves and an edge from u to v indicates that u is an ancestor of v. Weights on these edges indicate the phylogenetic distance. The phylogenetic diversity (PD) of a set of species A is the total weight of edges that are on any path between the root of the phylogenetic tree and a species in A. Selecting a small set of species that maximizes phylogenetic diversity for a given phylogenetic tree is an essential task in preservation planning, where limited resources naturally prevent saving all species. An optimal solution can be found with a greedy algorithm [Steel, Systematic Biology, 2005; Pardi and Goldman, PLoS Genetics, 2005]. However, when a food web representing predator-prey relationships is given, finding a set of species that optimizes phylogenetic diversity subject to the condition that each saved species should be able to find food among the preserved species is NP-hard [Spillner et al., IEEE/ACM, 2008]. We present a generalization of this problem, where, inspired by biological considerations, the food web has weighted edges to represent the importance of predator-prey relationships. We show that this version is NP-hard even when both structures, the food web and the phylogenetic tree, are stars. To cope with this intractability, we proceed in two directions. Firstly, we study special cases where a species can only survive if a given fraction of its prey is preserved. Secondly, we analyze these problems through the lens of parameterized complexity. Our results include that finding a solution is fixed-parameter tractable with respect to the vertex cover number of the food web, assuming the phylogenetic tree is a star.

Figures (7)

• Figure 1: (0): A hypothetical phylogenetic tree $\mathcal{T}$. For $A=\{x_3,x_7\}$ and $B=\{x_1,x_4,x_5\}$, blue edges are in $E_{\mathcal{T}}\xspace(A)$ and red edges are in $E_{\mathcal{T}}\xspace^+(B)$. For $i \in [3]$, ($i$) shows the $(A,B)$-contraction of $\mathcal{T}$ after Step $i$. To increase readability, edge weights are omitted.
• Figure 2: An illustration of the \ref{['lem:sub-solutions']}. Here, all edges are directed towards the right.
• Figure 3: In this figure, the complexity of rw-PDD and rw-PDD$_{\text{s}}$ with respect to several structural parameters of the food web is presented. The complexity of rw-PDD is in the top left of each box, and the complexity of rw-PDD$_{\text{s}}$ is in the bottom right. A parameter $p$ is marked in red () if rw-PDD / rw-PDD$_{\text{s}}$ is NP-hard for constant values of $p$, or in amber () or green () if rw-PDD$_{\text{s}}$ / rw-PDD$_{\text{s}}$ admits an XP-, or, respectively, an FPT-algorithm with respect to $p$. Classifying rw-PDD parameterized by distance to clique remains open. rw-PDD$_{\text{s}}$ with respect to treewidth is W[1]-hard twvssw and in XP. Two parameters $p_1$ and $p_2$ are connected with an edge if in every graph the parameter $p_1$ further up is bounded by a function in $p_2$. A more in-depth look into the hierarchy of graph parameters can be found in graphparameters.
• Figure 4: An illustrative example of how to compute the function $\Phi_M$ for values of ${\omega}\xspace(\rho v_i)$.
• Figure 5: An illustration of the transformation done to the food web to prove \ref{['cor:h-cluster']}. Black vertices are new.

Theorems & Definitions (27)

• Lemma 1
• Lemma 2
• Lemma 3: $\star$
• Theorem 4: $\star$
• Corollary 5
• Corollary 6: $\star$
• Lemma 7: $\star$
• Theorem 8: $\star$
• Theorem 9
• Corollary 10