PN Publications
Below you can find my publications and preprints on Phylogenetic Networks, with links and toggleable abstracts.
To filter by a certain topic, click one of the buttons.
2026
[9]
preprint
Orienting Unrooted Binary Networks Faster: Focus on the Generator
arXiv:2606.31597.
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 \(\mathcal{O}(5.3334^\ell + \ell)\) time and orienting all remaining edges in polynomial time, we solve the orientation problem in \(\mathcal{O}(5.3334^\ell \cdot n)\) time for tree-based networks and in \(\mathcal{O}(5.3334^\ell \cdot n^2)\) time for orchards, where \(n\) is the number of vertices of the graph. Extending this approach with further branching yields \(\mathcal{O}(10.6667^\ell \cdot n^2)\)-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 \(\mathcal{O}(12.2071^\ell \cdot n^2)\)-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.
@article{Schestag2026Orienting,
title = {{Orienting Unrooted Binary Networks Faster: Focus on the Generator}},
author = {Schestag, Jannik and Zeh, Norbert},
journal = {arXiv preprint},
year = {2026},
archivePrefix = {arXiv},
eprint = {2606.31597}
}
[8]
preprint
Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth
arXiv:2605.23319.
Identifying a subset of taxa that maximizes Phylogenetic Diversity (PD) is a cornerstone of quantitative conservation planning. Traditionally, PD is defined over a phylogenetic tree in which leaves represent present-day taxa and branch lengths capture the estimated evolutionary distinctiveness. While PD maximization is computationally tractable on trees with unit costs, the problem becomes NP-hard when transitioning to phylogenetic networks or to budgeted versions in which protecting taxa incurs non-homogeneous costs.
In this paper, we address these two challenges by providing definitions and a comprehensive analysis of three distinct variants of budgeted PD on networks. We conduct our study through the lens of a small structural parameter, node scanwidth (nsw), which measures the tree-likeness of a phylogenetic network. We show that two of the considered variants can be optimized in \(\mathcal{O}^*(2^{\mathsf{nsw}} \cdot B^2)\) time, where \(B\) is the budget. For the computationally harder, third variant, we provide an algorithm to compute PD scores in \(\mathcal{O}^*(3^{\mathsf{nsw}})\) time.
We further contribute the first exact algorithms to compute node scanwidth, recognizing that the utility of algorithms based on nsw depends on the ability to compute nsw and its corresponding decomposition. Our approaches integrate data reduction rules, dynamic programming, and an Integer Linear Programming formulation.
We validate our theoretical results through extensive experiments on highly reticulated, simulated networks containing several hundred taxa, using heterogeneous costs. Our implementation computes PD scores and optimal nsw in fractions of a second, even on the most challenging instances. Furthermore, our budgeted optimization algorithms significantly outperform existing benchmarks for computing PD on networks, which were previously limited to unit-cost scenarios. The software makes analyses even on networks with a thousand taxa tractable in practice.
In this paper, we address these two challenges by providing definitions and a comprehensive analysis of three distinct variants of budgeted PD on networks. We conduct our study through the lens of a small structural parameter, node scanwidth (nsw), which measures the tree-likeness of a phylogenetic network. We show that two of the considered variants can be optimized in \(\mathcal{O}^*(2^{\mathsf{nsw}} \cdot B^2)\) time, where \(B\) is the budget. For the computationally harder, third variant, we provide an algorithm to compute PD scores in \(\mathcal{O}^*(3^{\mathsf{nsw}})\) time.
We further contribute the first exact algorithms to compute node scanwidth, recognizing that the utility of algorithms based on nsw depends on the ability to compute nsw and its corresponding decomposition. Our approaches integrate data reduction rules, dynamic programming, and an Integer Linear Programming formulation.
We validate our theoretical results through extensive experiments on highly reticulated, simulated networks containing several hundred taxa, using heterogeneous costs. Our implementation computes PD scores and optimal nsw in fractions of a second, even on the most challenging instances. Furthermore, our budgeted optimization algorithms significantly outperform existing benchmarks for computing PD on networks, which were previously limited to unit-cost scenarios. The software makes analyses even on networks with a thousand taxa tractable in practice.
@article{holtgrefe2026tractable,
title = {{Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth}},
author = {Holtgrefe, Niels and Schestag, Jannik},
year = {2026},
journal = {arXiv preprint},
archivePrefix = {arXiv},
eprint = {2605.23319}
}
[7]
preprint
Average-Tree Phylogenetic Diversity Parameterized by Scanwidth and Invisibility
arXiv:2604.27745.
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 \(\mathcal{O}(2^{\mathrm{sw}} n)\) time, where \(\mathrm{sw}\) 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.
@article{vanIersel2026averagetree,
title = {{Average-Tree Phylogenetic Diversity Parameterized by Scanwidth and Invisibility}},
author = {van Iersel, Leo and Jones, Mark and Schestag, Jannik and Scornavacca, Celine and Weller, Mathias},
year = {2026},
archivePrefix = {arXiv},
journal = {arXiv preprint},
eprint = {2604.27745}
}
2025
[6]
preprint
PaNDA: Efficient Optimization of Phylogenetic Diversity in Networks
bioRxiv:10.1101/2025.11.14.688467.
Phylogenetic diversity plays an important role in biodiversity, conservation, and evolutionary studies by measuring the diversity of a set of taxa based on their phylogenetic relationships. In phylogenetic trees, a subset of \(k\) taxa with maximum phylogenetic diversity can be found by a simple and efficient greedy algorithm. However, this algorithmic tractability is lost when considering phylogenetic networks, which incorporate reticulate evolutionary events such as hybridization and horizontal gene transfer. To address this challenge, we introduce PaNDA (Phylogenetic Network Diversity Algorithms), the first software package and interactive graphical user-interface for exploring, visualizing and maximizing diversity in phylogenetic networks. PaNDA includes a novel algorithm to find a subset of \(k\) taxa with maximum diversity, running in polynomial time for networks of bounded canwidth, a measure of tree-likeness of a network that grows slower than the well-known level measure. This algorithm considers the variant of phylogenetic diversity on networks in which the branch lengths of all paths from the root to the selected taxa contribute towards their diversity. We demonstrate the scalability of this algorithm on simulated networks, successfully analyzing level-15 networks with up to 200 taxa in seconds. We also provide a proof-of-concept analysis using a phylogenetic network on Xiphophorus species, illustrating how the tool can support diversity studies based on real genomic data. The software is easily installable and freely available at https://github.com/nholtgrefe/panda. Additionally, we extend the definition of phylogene- tic diversity to semi-directed phylogenetic networks, which are mixed graphs increasingly used in phylogenetic analysis to model uncertainty of the root location. We prove that finding a subset of \(k\) taxa with maximum diversity remains NP-hard on semi-directed networks, but do present a polynomial-time algorithm for networks with bounded level.
@article{holtgrefe2025panda,
title = {{PaNDA: Efficient Optimization of Phylogenetic Diversity in Networks}},
author = {Holtgrefe, Niels and van Iersel, Leo and Meuwese, Ruben and Murakami, Yukihiro and Schestag, Jannik},
year = {2025},
publisher = {Cold Spring Harbor Laboratory},
journal = {bioRxiv: 10.1101/2025.11.14.688467},
doi = {10.1101/2025.11.14.688467}
}
[5]
conference paper
Parameterized Algorithms for Diversity of Networks with Ecological Dependencies
Proceedings of the 20th International Symposium on Parameterized and Exact Computation (IPEC 2025):11:1-11:21.
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 \(\overline{D}\), the scanwidth of the food web \(sw\), the maximum in-degree \(\delta\) 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.
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 \(\overline{D}\), the scanwidth of the food web \(sw\), the maximum in-degree \(\delta\) 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.
@inproceedings{jones2025parameterized,
title = {{Parameterized Algorithms for Diversity of Networks with Ecological Dependencies}},
author = {Jones, Mark and Schestag, Jannik},
booktitle = {Proceedings of the 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
pages = {11:1--11:21},
year = {2025},
organization = {Schloss-Dagstuhl-Leibniz Zentrum f{\"u}r Informatik},
doi = {10.4230/LIPIcs.IPEC.2025.11}
}
[4]
conference paper
Average-Tree Phylogenetic Diversity of Networks
Proceedings of the 25th International Workshop on Algorithms in Bioinformatics (WABI 2025):14:1--14:21.
Phylogenetic diversity is a measure used to quantify the biodiversity of a set of species. Here, we introduce the average-tree phylogenetic diversity score in rooted binary phylogenetic networks and consider algorithms for computing and maximizing the score on a given network. Basically, the score is the weighted average of the phylogenetic diversity scores in all trees displayed by the network, with the weights determined by the inheritance probabilities on the reticulation edges used in the embeddings. We show that computing the score of a given set of taxa in a given network is #P-hard, directly implying #P-hardness of finding a subset of \(k\) taxa achieving maximum diversity score and thereby ruling out polynomial-time algorithms for these problems unless the polynomial hierarchy collapses. However, we show that both problems can be solved efficiently if the input network is close to being a tree in the sense that its reticulation number is small. More precisely, we prove that we can solve the optimization problem in networks with \(n\) leaves and \(r\) reticulations in \(2^{\mathcal{O}(r)} \cdot n \cdot k\) time. Using experiments on data produced by simulating a reticulate-evolution process, we show that our algorithm runs efficiently on networks with hundreds of taxa and tens of reticulations.
@inproceedings{vanIersel2025averagetree,
title = {{Average-Tree Phylogenetic Diversity of Networks}},
author = {van Iersel, Leo and Jones, Mark and Schestag, Jannik and Scornavacca, Celine and Weller, Mathias},
booktitle = {Proceedings of the 25th International Workshop on Algorithms in Bioinformatics (WABI 2025)},
pages = {15:1--15:21},
year = {2025},
organization = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
doi = {10.4230/LIPIcs.WABI.2025.15}
}
[3]
conference paper
Phylogenetic Network Diversity Parameterized by Reticulation Number and Beyond
Proceedings of the 22nd RECOMB International Workshop on Comparative Genomics (RECOMB-CG 2025):107-130.
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 Maximize 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 \(\mathcal{O}(2^r \log(k)(n+r))\) time, with \(n\) the total number of species in the network and \(r\) its reticulation number. Furthermore, we show that Maximize 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.
@inproceedings{vanIersel2025phylogenetic,
title = {{Phylogenetic Network Diversity Parameterized by Reticulation Number and Beyond}},
author = {van Iersel, Leo and Jones, Mark and Schestag, Jannik and Scornavacca, Celine and Weller, Mathias},
booktitle = {Proceedings of the 22nd RECOMB International Workshop on Comparative Genomics (RECOMB-CG 2025)},
pages = {107--130},
year = {2025},
organization = {Springer},
doi = {10.1007/978-3-031-94928-9_7}
}
2023
[2]
conference paper
Finding Degree-Constrained Acyclic Orientations
Proceedings of the 18th International Symposium on Parameterized and Exact Computation (IPEC 2023):19:1--19:14.
We consider the problem of orienting a given, undirected graph into a (directed) acyclic graph such that the in-degree of each vertex \(v\) is in a prescribed list \(\lambda(v)\). Variants of this problem have been studied for a long time and with various applications, but mostly without the requirement for acyclicity. Without this requirement, the problem is closely related to the classical General Factor problem, which is known to be NP-hard in general, but polynomial-time solvable if no list \(\lambda(v)\) contains large “gaps” [Cornuéjols, J. Comb. Theory B, 1988]. In contrast, we show that deciding if an acyclic orientation exists is NP-hard even in the absence of such “gaps”.
On the positive side, we design parameterized algorithms for various, natural parameterizations of the acyclic orientation problem. A special case of the orientation problem with degree constraints recently came up in the context of reconstructing evolutionary histories (that is, phylogenetic networks). This phylogenetic setting imposes additional structure onto the problem that can be exploited algorithmically, allowing us to show fixed-parameter tractability when parameterized by either the treewidth of \(G\) (a smaller parameter than the frequently employed “level”), by the number of vertices \(v\) for which \(|\lambda(v)| \geq 2\), by the number of vertices \(v\) for which the highest value in \(\lambda(v)\) is at least 2. While the latter result can be extended to the general degree-constraint acyclic orientation problem, we show that the former cannot unless FPT=W[1].
On the positive side, we design parameterized algorithms for various, natural parameterizations of the acyclic orientation problem. A special case of the orientation problem with degree constraints recently came up in the context of reconstructing evolutionary histories (that is, phylogenetic networks). This phylogenetic setting imposes additional structure onto the problem that can be exploited algorithmically, allowing us to show fixed-parameter tractability when parameterized by either the treewidth of \(G\) (a smaller parameter than the frequently employed “level”), by the number of vertices \(v\) for which \(|\lambda(v)| \geq 2\), by the number of vertices \(v\) for which the highest value in \(\lambda(v)\) is at least 2. While the latter result can be extended to the general degree-constraint acyclic orientation problem, we show that the former cannot unless FPT=W[1].
@inproceedings{garvardt2023finding,
title = {{Finding Degree-Constrained Acyclic Orientations}},
author = {Garvardt, Jaroslav and Renken, Malte and Schestag, Jannik and Weller, Mathias},
booktitle = {Proceedings of the 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
pages = {19:1--19:14},
year = {2023},
organization = {Schloss-Dagstuhl-Leibniz Zentrum f{\"u}r Informatik},
doi = {10.4230/LIPIcs.IPEC.2023.19}
}
[1]
conference paper
Maximizing All-Paths Phylogenetic Diversity: Parameterized Approaches for Networks
Proceedings of the 18th International Symposium on Parameterized and Exact Computation (IPEC 2023):30:1-30:12.
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.
@inproceedings{jones2023maximize,
title = {{How Can We Maximize Phylogenetic Diversity? Parameterized Approaches for Networks}},
author = {Jones, Mark and Schestag, Jannik},
booktitle = {Proceedings of the 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
pages = {30:1--30:12},
year = {2023},
organization = {Schloss-Dagstuhl-Leibniz Zentrum f{\"u}r Informatik},
doi = {10.4230/LIPIcs.IPEC.2023.30}
}