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Parameterized Algorithms for Diversity of Networks with Ecological Dependencies

Mark Jones, Jannik Schestag

TL;DR

This work studies the problem of selecting a size-$k$ taxon set on phylogenetic networks to maximize All-Paths phylogenetic diversity $PD_{\mathcal{N}}(A)$ under viability constraints derived from a food web $\mathcal{F}$. It introduces Map-Weighted-PDD and its special cases ($\varepsilon$-PDD, $1$-PDD), and analyzes their parameterized complexity with respect to $k$, $\overline{D}=PD_{\mathcal{N}}(X)-D$, $sw_{\mathcal{F}}$, $\delta$, and $h$, providing a complete dichotomy: several cases are fixed-parameter tractable (FPT) while others are W[1]-hard when parameters are restricted. A central methodological contribution is a novel color-coding framework using a single color per edge and the concept of perfect triples, enabling FPT algorithms for complex dependencies induced by ecological constraints; this framework also encompasses an anchor-based approach with a colored subproblem ex-$N$-colored-Map-W-PDD. The results show FPT algorithms for Map-Weighted-PDD under $\overline{D}+sw_{\mathcal{F}}$ and for $k+sw_{\mathcal{F}}+\delta+h$, while identifying hardness when parameters are reduced, and they outline open questions (notably a potential FPT status for Map-$\varepsilon$-PDD with $k+\delta+h$). Overall, the paper advances computational biodiversity conservation by providing rigorous parameterized algorithms for diversity under ecological dependencies and clarifying the tractability frontier for these biologically motivated problems.

Abstract

For a phylogenetic tree, the phylogenetic diversity of a set A of taxa is the total weight of edges on paths to A. Finding small sets of maximal diversity is crucial for conservation planning, as it indicates where limited resources can be invested most efficiently. In recent years, efficient algorithms have been developed to find sets of taxa that maximize phylogenetic diversity either in a phylogenetic network or in a phylogenetic tree subject to ecological constraints, such as a food web. However, these aspects have mostly been studied independently. Since both factors are biologically important, it seems natural to consider them together. In this paper, we introduce decision problems where, given a phylogenetic network, a food web, and integers k, and D, the task is to find a set of k taxa with phylogenetic diversity of at least D under the maximize all paths measure, while also satisfying viability conditions within the food web. Here, we consider different definitions of viability, which all demand that a "sufficient" number of prey species survive to support surviving predators. We investigate the parameterized complexity of these problems and present several fixed-parameter tractable (FPT) algorithms. Specifically, we provide a complete complexity dichotomy characterizing which combinations of parameters - out of the size constraint k, the acceptable diversity loss D, the scanwidth of the food web, the maximum in-degree in the network, and the network height h - lead to W[1]-hardness and which admit FPT algorithms. Our primary methodological contribution is a novel algorithmic framework for solving phylogenetic diversity problems in networks where dependencies (such as those from a food web) impose an order, using a color coding approach.

Parameterized Algorithms for Diversity of Networks with Ecological Dependencies

TL;DR

This work studies the problem of selecting a size- taxon set on phylogenetic networks to maximize All-Paths phylogenetic diversity under viability constraints derived from a food web . It introduces Map-Weighted-PDD and its special cases (-PDD, -PDD), and analyzes their parameterized complexity with respect to , , , , and , providing a complete dichotomy: several cases are fixed-parameter tractable (FPT) while others are W[1]-hard when parameters are restricted. A central methodological contribution is a novel color-coding framework using a single color per edge and the concept of perfect triples, enabling FPT algorithms for complex dependencies induced by ecological constraints; this framework also encompasses an anchor-based approach with a colored subproblem ex--colored-Map-W-PDD. The results show FPT algorithms for Map-Weighted-PDD under and for , while identifying hardness when parameters are reduced, and they outline open questions (notably a potential FPT status for Map--PDD with ). Overall, the paper advances computational biodiversity conservation by providing rigorous parameterized algorithms for diversity under ecological dependencies and clarifying the tractability frontier for these biologically motivated problems.

Abstract

For a phylogenetic tree, the phylogenetic diversity of a set A of taxa is the total weight of edges on paths to A. Finding small sets of maximal diversity is crucial for conservation planning, as it indicates where limited resources can be invested most efficiently. In recent years, efficient algorithms have been developed to find sets of taxa that maximize phylogenetic diversity either in a phylogenetic network or in a phylogenetic tree subject to ecological constraints, such as a food web. However, these aspects have mostly been studied independently. Since both factors are biologically important, it seems natural to consider them together. In this paper, we introduce decision problems where, given a phylogenetic network, a food web, and integers k, and D, the task is to find a set of k taxa with phylogenetic diversity of at least D under the maximize all paths measure, while also satisfying viability conditions within the food web. Here, we consider different definitions of viability, which all demand that a "sufficient" number of prey species survive to support surviving predators. We investigate the parameterized complexity of these problems and present several fixed-parameter tractable (FPT) algorithms. Specifically, we provide a complete complexity dichotomy characterizing which combinations of parameters - out of the size constraint k, the acceptable diversity loss D, the scanwidth of the food web, the maximum in-degree in the network, and the network height h - lead to W[1]-hardness and which admit FPT algorithms. Our primary methodological contribution is a novel algorithmic framework for solving phylogenetic diversity problems in networks where dependencies (such as those from a food web) impose an order, using a color coding approach.

Paper Structure

This paper contains 18 sections, 24 theorems, 1 equation, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Related work
  5. Preliminary Observations
  6. Parameter ${{\overline{D}}\xspace + {\mathop{\mathrm{sw}}\nolimits_{{\mathcal{F}}\xspace}}\xspace}$
  7. Parameter ${k + {\mathop{\mathrm{sw}}\nolimits_{{\mathcal{F}}\xspace}}\xspace + \delta + h}$
  8. Map-${\varepsilon}$-PDD with respect to ${k+\delta+h}$
  9. Discussion
  10. Appendix---Omitted proofs
  11. Proof of \ref{['thm:Dbar+sw']}
  12. Proof of \ref{['lem:perfectTripleBound']}
  13. Proof of \ref{['lem:exPPD']}
  14. Proof of \ref{['lem:Sbar']}
  15. Proof of \ref{['lem:fkSMAPPD']}
  16. ...and 3 more sections

Key Result

Proposition 2

For any integers $n, k \geq 1$, a $(n,k)$-perfect hash family of size $e^k k^{\mathcal{O}(\log k)} \cdot \log n$ can be constructed in time $e^k k^{\mathcal{O}(\log k)} \cdot n \log n$.

Figures (4)

  • Figure 1: Top Left: An example network. The set $F_{x,C}$ is highlighted in and one possible set $F_{x,C}'$ in . For the sake of readability, colors of edges in $F_{x,C}$ are omitted. Bottom Left: The bipartite graph which is constructed in \ref{['lem:FxC-compute']}. A perfect matching is highlighted. Right: Four examples of networks, where $F_{x,C}$ is not a respecting set of edges.
  • Figure 2: A hypothetical network on 3 colors $\{,,\}$. For each taxon the $(x,\chi)$-suitable set $F_{x,\chi}$ is indicated in , , or . In this network, ${H}\xspace$ takes the value of 3---there are three tree edges on paths to $x_3$ or $x_4$.
  • Figure 3: An example for \ref{['lem:Sbar']} where ${H}\xspace$ is maximized. This is the case if above the leaf an upside-down pyramid of reticulations is followed by $\delta^{h_r}$ paths of tree vertices of length $h_t$.
  • Figure 4: An execution of Reduction Rule 6 in PDD. Left is the instance before the execution and right is the instance after the execution. In both instances, to the left is the pattern tree and on the right the phylogenetic tree.

Theorems & Definitions (28)

  • Definition 1
  • Proposition 2: cyganNaor1995SplittersAN
  • Theorem 3: faller
  • Theorem 4
  • Theorem 5: bordewichNetworksMAPPD
  • Corollary 6
  • Corollary 7
  • Theorem 8: $\star$
  • Corollary 9
  • Definition 10
  • ...and 18 more