arXiv subjects

Jannik Schestag

Publications and source records attributed to Jannik Schestag.

15 recordsLinked to original sources

arXiv2026

Maximizing All-Paths Phylogenetic Diversity: Parameterized Approaches for Networks

Phylogenetic Diversity (PD) is a fundamental measure of biodiversity, originally defined on phylogenetic trees and widely used in conservation biology. Phylogenetic trees are often generalised to directed acyclic graphs, called phylogenetic networks. As such, a corresponding generalization of PD is needed. A natural generalization to edge-weighted phylogenetic networks is the all-paths measure, where the diversity of a set S of species (taxa) is defined as the total weight of all edges that lie on a path from the root to at least one species in S. While maximizing PD on trees can be solved in polynomial time, the corresponding problem on networks is NP-hard and difficult to approximate. We undertake a systematic parameterized complexity study of the Max-All-Paths-PD (MapPD) problem. We establish W[2]-hardness when parameterized by the number of species that are included in a solution, and W[1]-hardness for the number of species that are excluded. On the positive side, we show that the problem is fixed-parameter tractable with respect to the threshold of diversity and the acceptable loss of diversity. We further analyze how the network's proximity to a tree influences algorithmic behavior and present single-exponential fixed-parameter algorithms when parameterized by the number of reticulations and by the treewidth of the underlying graph. Finally, we present a polynomial kernelization for MapPD with respect to the number of reticulation edges.

cs.DS↗

arXiv2026

Orienting Unrooted Binary Networks Faster: Focus on the Generator

The problem of orienting an unrooted network to obtain a specific class of rooted phylogenetic networks is known to be NP-hard in many cases. In this paper, we introduce two algorithmic frameworks that yield significantly improved fixed-parameter tractable (FPT) algorithms parameterized by the network level $\ell$. Our first main contribution shows that for several prominent network classes, the core algorithmic difficulty lies in finding a directed spanning tree on the network's undirected generator. By enumerating these spanning trees in $O(5.3334^\ell + \ell)$ time and orienting all remaining edges in polynomial time, we solve the orientation problem in $O(5.3334^\ell \cdot n)$ time for tree-based networks and in $O(5.3334^\ell \cdot n^2)$ time for orchards, where $n$ is the number of vertices of the graph. Extending this approach with further branching yields $O(10.6667^\ell \cdot n^2)$-time algorithms for tree-child and normal networks. Our second technique bypasses spanning trees by directly guessing the placement of reticulations on the generator. This framework provides $O(12.2071^\ell \cdot n^2)$-time algorithms for temporal, reticulation-visible, and tree-sibling networks. Finally, we demonstrate the versatility of the reticulation-guessing framework by showing that even computing an orientation with minimum scanwidth is single-exponential FPT with respect to the level. Together, these results significantly improve the best-known running times for phylogenetic network orientation.

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arXiv2026

Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth

Background: Identifying a subset of taxa that maximizes phylogenetic diversity is a cornerstone of quantitative conservation planning. Traditionally, phylogenetic diversity is defined over a phylogenetic tree in which leaves resemble present-day taxa and the branch lengths capture the estimated evolutionary distinctiveness. While maximizing phylogenetic diversity is computationally tractable on trees with unit costs, the problem becomes computationally intractable when transitioning to phylogenetic networks or to budgeted versions in which protecting taxa incurs non-homogeneous costs. This paper addresses these two challenges together, providing definitions and a comprehensive analysis of three distinct variants of budgeted phylogenetic diversity on networks.Results: We conduct our study through the lens of a small structural parameter, node scanwidth ($nsw$), which measures the tree-likeness of a phylogenetic network. Given a tree-extension of width $nsw$, we show that two of the considered variants can be optimized in $\mathcal{O}^*(3^{nsw} \cdot B^2)$ time, where $B$ is the budget. For the computationally harder third variant, we provide an algorithm to compute phylogenetic diversity scores in $\mathcal{O}^*(4^{nsw})$ time. We further contribute the first exact algorithms to compute node scanwidth itself. On highly reticulated, simulated networks with several hundred taxa and heterogeneous costs, our implementation computes phylogenetic diversity scores and optimal node scanwidth in fractions of a second, and the budgeted optimization algorithms significantly outperform existing benchmarks previously limited to unit-cost scenarios. Conclusions: Node scanwidth proves to be a small and computable parameter that makes budgeted phylogenetic diversity tractable on realistic networks, scaling comfortably to a thousand taxa. This narrows the gap between reali...

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arXiv2026

Average-Tree Phylogenetic Diversity Parameterized by Scanwidth and Invisibility

We investigate parameterized algorithms for computing the average-tree phylogenetic diversity (APD) in rooted phylogenetic networks, studying the problem under different structural parameters that capture the deviation of a network from a tree. Our primary parameter is the scanwidth, a measure of the tree-likeness of a given directed acyclic graph. We show that a subset of taxa with maximum APD can be found in polynomial time in phylogenetic networks of scanwidth at most 2, but becomes NP-hard in networks of scanwidth 3. Further, we design an algorithm that computes the APD of a given set of taxa in O(2^sw n) time, where sw denotes the scanwidth and n the number of taxa in the input network. Finally, we give a linear-time algorithm for computing the APD of a given set of taxa if the network induced by these taxa is reticulation-visible. We generalize this algorithm to still run in polynomial time if each biconnected component of the induced network has only constantly many invisible reticulations.

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arXiv2026

Limits of Kernelization and Parametrization for Phylogenetic Diversity with Dependencies

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[...]

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arXiv2026

The First Known Problem That Is FPT with Respect to Node Scanwidth but Not Treewidth

Structural parameters of graphs, such as treewidth, play a central role in the study of the parameterized complexity of graph problems. Motivated by the study of parametrized algorithms on phylogenetic networks, scanwidth was introduced recently as a new treewidth-like structural parameter for directed acyclic graphs (DAGs) that respects the edge directions in the DAG. The utility of this width measure has been demonstrated by results that show that a number of problems that are fixed-parameter tractable (FPT) with respect to both treewidth and scanwidth allow algorithms with a better dependence on scanwidth than on treewidth. More importantly, these scanwidth-based algorithms are often much simpler than their treewidth-based counterparts: the name ``scanwidth'' reflects that traversing a tree extension (the scanwidth-equivalent of a tree decomposition) of a DAG amounts to ``scanning'' the DAG according to a well-chosen topological ordering. While these results show that scanwidth is useful especially for solving problems on phylogenetic networks, all problems studied through the lens of scanwidth so far are either FPT with respect to both scanwidth and treewidth, or W[$\ell$]-hard, for some $\ell \ge 1$, with respect to both. In this paper, we show that scanwidth is not just a proxy for treewidth and provides information about the structure of the input graph not provided by treewidth, by proving a fairly stark complexity-theoretic separation between these two width measures. Specifically, we prove that Weighted Phylogenetic Diversity with Dependencies is FPT with respect to the scanwidth of the food web but W[$\ell$]-hard with respect to its treewidth, for all $\ell \ge 1$. To the best of our knowledge, no such separation between these two width measures has been shown for any problem before.

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arXiv2025

Parameterized Algorithms for Diversity of Networks with Ecological Dependencies

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.

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arXiv2025

Weighted Food Webs Make Computing Phylogenetic Diversity So Much Harder

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.

q-bio.PE↗

arXiv2025

Who Should Have a Place on the Ark? Parameterized Algorithms for the Maximization of Phylogenetic Diversity

Phylogenetic Diversity(PD)is a well-regarded measure of the overall biodiversity of a set of present-day species(taxa)that indicates its ecological significance.In the Maximize Phylogenetic Diversity(Max-PD)problem one is asked to find a small set of taxa in a phylogenetic tree for which this measure is maximized.Max-PD is particularly relevant in conservation planning,where limited resources necessitate prioritizing certain taxa to minimize biodiversity loss.Although Max-PD can be solved in polynomial time [Steel,SB,2005;Pardi&Goldman,PLoS,2005],its generalizations-which aim to model biological processes and other aspects in conservation planning with greater accuracy-often exhibit NP-hardness,making them computationally challenging.This thesis explores a selection of these generalized problems within the framework of parameterized complexity. In Generalized Noah's Ark Problem(GNAP),each taxon only survives at a certain survival probability,which can be increased by investing more money in the taxon.We show that GNAP is W[1]-hard with respect to the number of taxa but is XP with respect to the number of different costs and different survival probabilities. Additionally,we show that unit-cost-NAP,a special case of GNAP,is NP-hard. In Time Sensitive Maximization of Phylogenetic Diversity(Time-PD),different extinction times of taxa are considered after which they can no longer be saved.For Time-PD,we present color-coding algorithms that prove that Time-PD is fixed-parameter tractable(FPT)with respect to the threshold of diversity and the acceptable loss of diversity. In Optimizing PD with Dependencies(PDD),each saved taxon must be a source in the ecological system or a predator of another saved species.These dependencies are given in a food-web.We show that PDD is FPT when parameterized with the size of the solution plus the height of the phylogenetic tree. Further,we consider pa...

cs.DM↗

arXiv2024

Maximizing Phylogenetic Diversity under Ecological Constraints: A Parameterized Complexity Study

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.

cs.CC↗

arXiv2024

Phylogenetic Network Diversity Parameterized by Reticulation Number and Beyond

Network Phylogenetic Diversity (Network-PD) is a measure for the diversity of a set of species based on a rooted phylogenetic network (with branch lengths and inheritance probabilities on the reticulation edges) describing the evolution of those species. We consider the Max-Network-PD problem: Given such a network, find k species with maximum Network-PD score. We show that this problem is fixed-parameter tractable (FPT) for binary networks, by describing an optimal algorithm running in O(2r log(k)(n + r)) time, with n the total number of species in the network and r its reticulation number. Furthermore, we show that Max-Network-PD is NP-hard for level-1 networks, proving that, unless P=NP, the FPT approach cannot be extended by using the level as parameter instead of the reticulation number.

cs.CC↗

arXiv2024

Maximizing Phylogenetic Diversity under Time Pressure: Planning with Extinctions Ahead

Phylogenetic Diversity (PD) is a measure of the overall biodiversity of a set of present-day species (taxa) within a phylogenetic tree. In Maximize Phylogenetic Diversity (MPD) one is asked to find a set of taxa (of bounded size/cost) for which this measure is maximized. MPD is a relevant problem in conservation planning, where there are not enough resources to preserve all taxa and minimizing the overall loss of biodiversity is critical. We consider an extension of this problem, motivated by real-world concerns, in which each taxon not only requires a certain amount of time to save, but also has an extinction time after which it can no longer be saved. In addition there may be multiple teams available to work on preservation efforts in parallel; we consider two variants of the problem based on whether teams are allowed to collaborate on the same taxa. These problems have much in common with machine scheduling problems, (with taxa corresponding to tasks and teams corresponding to machines), but with the objective function (the phylogenetic diversity) inspired by biological considerations. Our extensions are, in contrast to the original MPD, NP-hard, even in very restricted cases. We provide several algorithms and hardness-results and thereby show that the problems are fixed-parameter tractable (FPT) when parameterized the target phylogenetic diversity, and that the problem where teams are allowed to collaborate is FPT when parameterized the acceptable loss of diversity.

cs.CC↗

arXiv2024

Finding a Sparse Connected Spanning Subgraph in a non-Uniform Failure Model

We study a generalization of the classic Spanning Tree problem that allows for a non-uniform failure model. More precisely, edges are either \emph{safe} or \emph{unsafe} and we assume that failures only affect unsafe edges. In Unweighted Flexible Graph Connectivity we are given an undirected graph $G = (V,E)$ in which the edge set $E$ is partitioned into a set $S$ of safe edges and a set $U$ of unsafe edges and the task is to find a set $T$ of at most $k$ edges such that $T - \{u\}$ is connected and spans $V$ for any unsafe edge $u \in T$. Unweighted Flexible Graph Connectivity generalizes both Spanning Tree and Hamiltonian Cycle. We study Unweighted Flexible Graph Connectivity in terms of fixed-parameter tractability (FPT). We show an almost complete dichotomy on which parameters lead to fixed-parameter tractability and which lead to hardness. To this end, we obtain FPT-time algorithms with respect to the vertex deletion distance to cluster graphs and with respect to the treewidth. By exploiting the close relationship to Hamiltonian Cycle, we show that FPT-time algorithms for many smaller parameters are unlikely under standard parameterized complexity assumptions. Regarding problem-specific parameters, we observe that Unweighted Flexible Graph Connectivity} admits an FPT-time algorithm when parameterized by the number of unsafe edges. Furthermore, we investigate a below-upper-bound parameter for the number of edges of a solution. We show that this parameter also leads to an FPT-time algorithm.

cs.DM↗

arXiv2023

A Multivariate Complexity Analysis of the Generalized Noah's Ark Problem

In the Generalized Noah's Ark Problem, one is given a phylogenetic tree on a set of species X and a set of conservation projects for each species. Each project comes with a cost and raises the survival probability of the corresponding species. The aim is to select a conservation project for each species such that the total cost of the selected projects does not exceed some given threshold and the expected phylogenetic diversity is as large as possible. We study the complexity of Generalized Noah's Ark Problem and some of its special cases with respect to several parameters related to the input structure, such as the number of different costs, the number of different survival probabilities, or the number of species, |X|.

cs.DS↗

arXiv2021

Destroying Bicolored $P_3$s by Deleting Few Edges

We introduce and study the Bicolored $P_3$ Deletion problem defined as follows. The input is a graph $G=(V,E)$ where the edge set $E$ is partitioned into a set $E_r$ of red edges and a set $E_b$ of blue edges. The question is whether we can delete at most $k$ edges such that $G$ does not contain a bicolored $P_3$ as an induced subgraph. Here, a bicolored $P_3$ is a path on three vertices with one blue and one red edge. We show that Bicolored $P_3$ Deletion is NP-hard and cannot be solved in $2^{o(|V|+|E|)}$ time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored $P_3$ Deletion is polynomial-time solvable when $G$ does not contain a bicolored $K_3$, that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case that $G$ contains no blue $P_3$, red $P_3$, blue $K_3$, and red $K_3$. Finally, we show that Bicolored $P_3$ Deletion can be solved in $ O(1.84^k\cdot |V| \cdot |E|)$ time and that it admits a kernel with $ O(kΔ\min(k,Δ))$ vertices, where $Δ$ is the maximum degree of $G$.

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