arXiv subjects

Mathias Weller

Publications and source records attributed to Mathias Weller.

16 recordsLinked to original sources

arXiv2026

Tree Containment Parameterized by Scanwidth

TREE CONTAINMENT is a central decision problem in mathematical phylogenetics, asking whether a given rooted phylogenetic tree is embeddable in ("displayed by") a given rooted phylogenetic network. While the problem is NP-complete for general networks, many algorithmic advances have relied on structural parameters that capture how "tree-like" a network is. In this paper we investigate TREE CONTAINMENT under the structural parameter scanwidth, a directed width measure generalizing popular parameters measuring tree-likeness of phylogenetic networks. We first present a parameterized algorithm that solves the problem in $O(4^{k + k\log{k}} n + nm^2)$ time, where $n$ and $m$ are the numbers of nodes and arcs in the network and $k$ is the width of a given tree-extension. Complementing this upper bound, we prove a matching lower bound under the Exponential-Time Hypothesis (ETH), showing that there is no algorithm for TREE CONTAINMENT that runs in $2^{o(c\log{c})} n^{O(1)}$ time, even on binary inputs, where $c$ is the directed cutwidth of the input network, which upper-bounds the scanwidth $k$.

cs.DS↗

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.

cs.DS↗

arXiv2025

Exploiting Low Scanwidth to Resolve Soft Polytomies

Phylogenetic networks allow modeling reticulate evolution, capturing events such as hybridization and horizontal gene transfer. A fundamental computational problem in this context is the Tree Containment problem, which asks whether a given phylogenetic network is compatible with a given phylogenetic tree. However, the classical statement of the problem is not robust to poorly supported branches in biological data, possibly leading to false negatives. In an effort to address this, a relaxed version that accounts for uncertainty, called Soft Tree Containment, has been introduced by Bentert, Mal\'ik, and Weller [SWAT'18]. We present an algorithm that solves Soft Tree Containment in $2^{O(\Delta_T \cdot k \cdot \log(k))} \cdot n^{O(1)}$ time, where $k = \operatorname{sw}(\Gamma) + \Delta_N$, with $\Delta_T$ and $\Delta_N$ denoting the maximum out-degrees in the tree and the network, respectively, and $\operatorname{sw}(\Gamma)$ denoting the "scanwidth" [Berry, Scornavacca, and Weller, SOFSEM'20] of a given tree extension of the network, while $n$ is the input size. Our approach leverages the fact that phylogenetic networks encountered in practice often exhibit low scanwidth, making the problem more tractable.

cs.DS↗

arXiv2025

When Many Trees Go to War: On Sets of Phylogenetic Trees With Almost No Common Structure

It is known that any two trees on the same $n$ leaves can be displayed by a network with $n-2$ reticulations, and there are two trees that cannot be displayed by a network with fewer reticulations. But how many reticulations are needed to display multiple trees? For any set of $t$ trees on $n$ leaves, there is a trivial network with $(t - 1)n$ reticulations that displays them. To do better, we have to exploit common structure of the trees to embed non-trivial subtrees of different trees into the same part of the network. In this paper, we show that for $t \in o(\sqrt{\lg n})$, there is a set of $t$ trees with virtually no common structure that could be exploited. More precisely, we show for any $t\in o(\sqrt{\lg n})$, there are $t$ trees such that any network displaying them has $(t-1)n - o(n)$ reticulations. For $t \in o(\lg n)$, we obtain a slightly weaker bound. We also prove that already for $t = c\lg n$, for any constant $c > 0$, there is a set of $t$ trees that cannot be displayed by a network with $o(n \lg n)$ reticulations, matching up to constant factors the known upper bound of $O(n \lg n)$ reticulations sufficient to display \emph{all} trees with $n$ leaves. These results are based on simple counting arguments and extend to unrooted networks and trees.

math.CO↗

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↗

arXiv2023

Neighbor Joining And Leaf Status

The Neighbor Joining Algorithm is among the most fundamental algorithmic results in computational biology. However, its definition and correctness proof are not straightforward. In particular, ''the question ''what does the NJ method seek to do?'' has until recently proved somewhat elusive'' [Gascuel \& Steel, 2006]. While a rigorous mathematical analysis is now available, it is still considered somewhat hard to follow and its proof tedious at best. In this work, we present an alternative interpretation of the goal of the Neighbor Joining algorithm by proving that it chooses to merge the two taxa u and v that maximize the ''leaf-status'', that is, the sum of distances of all leaves to the unique u-v-path.

q-bio.QM↗

arXiv2022

Embedding phylogenetic trees in networks of low treewidth

Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called \textsc{Tree Containment}, arises when validating networks constructed by phylogenetic inference methods.We present the first algorithm for (rooted) \textsc{Tree Containment} using the treewidth $t$ of the input network $N$ as parameter, showing that the problem can be solved in $2^{O(t^2)}\cdot|N|$ time and space.

q-bio.PE↗

arXiv2019

Listing Conflicting Triples in Optimal Time

Different sources of information might tell different stories about the evolutionary history of a given set of species. This leads to (rooted) phylogenetic trees that "disagree" on triples of species, which we call "conflict triples". An important subtask of computing consensus trees which is interesting in its own regard is the enumeration of all conflicts exhibited by a pair of phylogenetic trees (on the same set of $n$ taxa). As it is possible that a significant part of the $n^3$ triples are in conflict, the trivial ${\Theta}(n^3)$-time algorithm that checks for each triple whether it constitutes a conflict, was considered optimal. It turns out, however, that we can do way better in the case that there are only few conflicts. In particular, we show that we can enumerate all d conflict triples between a pair of phylogenetic trees in $O(n + d)$ time. Since any deterministic algorithm has to spend ${\Theta}(n)$ time reading the input and ${\Theta}(d)$ time writing the output, no deterministic algorithm can solve this task faster than we do (up to constant factors).

cs.DS↗

arXiv2019

Fast Exact Dynamic Time Warping on Run-Length Encoded Time Series

Dynamic Time Warping (DTW) is a well-known similarity measure for time series. The standard dynamic programming approach to compute the DTW distance of two length-$n$ time series, however, requires~$O(n^2)$ time, which is often too slow for real-world applications. Therefore, many heuristics have been proposed to speed up the DTW computation. These are often based on lower bounding techniques, approximating the DTW distance, or considering special input data such as binary or piecewise constant time series. In this paper, we present a first exact algorithm to compute the DTW distance of two run-length encoded time series whose running time only depends on the encoding lengths of the inputs. The worst-case running time is cubic in the encoding length. In experiments we show that our algorithm is indeed fast for time series with short encoding lengths.

cs.DS↗

arXiv2018

What is known about Vertex Cover Kernelization?

We are pleased to dedicate this survey on kernelization of the Vertex Cover problem, to Professor Juraj Hromkovi\v{c} on the occasion of his 60th birthday. The Vertex Cover problem is often referred to as the Drosophila of parameterized complexity. It enjoys a long history. New and worthy perspectives will always be demonstrated first with concrete results here. This survey discusses several research directions in Vertex Cover kernelization. The Barrier Degree of Vertex Cover kernelization is discussed. We have reduction rules that kernelize vertices of small degree, including in this paper new results that reduce graphs almost to minimum degree five. Can this process go on forever? What is the minimum vertex-degree barrier for polynomial-time kernelization? Assuming the Exponential-Time Hypothesis, there is a minimum degree barrier. The idea of automated kernelization is discussed. We here report the first experimental results of an AI-guided branching algorithm for Vertex Cover whose logic seems amenable for application in finding reduction rules to kernelize small-degree vertices. The survey highlights a central open problem in parameterized complexity. Happy Birthday, Juraj!

cs.DS↗

arXiv2017

Constructing a Consensus Phylogeny from a Leaf-Removal Distance

Understanding the evolution of a set of genes or species is a fundamental problem in evolutionary biology. The problem we study here takes as input a set of trees describing {possibly discordant} evolutionary scenarios for a given set of genes or species, and aims at finding a single tree that minimizes the leaf-removal distance to the input trees. This problem is a specific instance of the general consensus/supertree problem, widely used to combine or summarize discordant evolutionary trees. The problem we introduce is specifically tailored to address the case of discrepancies between the input trees due to the misplacement of individual taxa. Most supertree or consensus tree problems are computationally intractable, and we show that the problem we introduce is also NP-hard. We provide tractability results in form of a 2-approximation algorithm. We also introduce a variant that minimizes the maximum number $d$ of leaves that are removed from any input tree, and provide a parameterized algorithm for this problem with parameter $d$.

cs.DS↗

arXiv2017

Linear-Time Tree Containment in Phylogenetic Networks

We consider the NP-hard Tree Containment problem that has important applications in phylogenetics. The problem asks if a given leaf-labeled network contains a subdivision of a given leaf-labeled tree. We develop a fast algorithm for the case that the input network is indeed a tree in which multiple leaves might share a label. By combining this algorithm with a generalization of a previously known decomposition scheme, we improve the running time on reticulation visible networks and nearly stable networks to linear time. While these are special classes of networks, they rank among the most general of the previously considered classes.

cs.CC↗

arXiv2015

On the Complexity of Hub Labeling

Hub Labeling (HL) is a data structure for distance oracles. Hierarchical HL (HHL) is a special type of HL, that received a lot of attention from a practical point of view. However, theoretical questions such as NP-hardness and approximation guarantee for HHL algorithms have been left aside. In this paper we study HL and HHL from the complexity theory point of view. We prove that both HL and HHL are NP-hard, and present upper and lower bounds for the approximation ratios of greedy HHL algorithms used in practice. We also introduce a new variant of the greedy HHL algorithm and a proof that it produces small labels for graphs with small highway dimension.

cs.DS↗

arXiv2014

Optimal Hub Labeling is NP-complete

Distance labeling is a preprocessing technique introduced by Peleg [Journal of Graph Theory, 33(3)] to speed up distance queries in large networks. Herein, each vertex receives a (short) label and, the distance between two vertices can be inferred from their two labels. One such preprocessing problem occurs in the hub labeling algorithm [Abraham et al., SODA'10]: the label of a vertex v is a set of vertices x (the "hubs") with their distance d(x,v) to v and the distance between any two vertices u and v is the sum of their distances to a common hub. The problem of assigning as few such hubs as possible was conjectured to be NP-hard, but no proof was known to date. We give a reduction from the well-known Vertex Cover problem on graphs to prove that finding an optimal hub labeling is indeed NP-hard.

cs.CC↗

arXiv2014

A Polynomial-time Algorithm for Outerplanar Diameter Improvement

The Outerplanar Diameter Improvement problem asks, given a graph $G$ and an integer $D$, whether it is possible to add edges to $G$ in a way that the resulting graph is outerplanar and has diameter at most $D$. We provide a dynamic programming algorithm that solves this problem in polynomial time. Outerplanar Diameter Improvement demonstrates several structural analogues to the celebrated and challenging Planar Diameter Improvement problem, where the resulting graph should, instead, be planar. The complexity status of this latter problem is open.

cs.DS↗

arXiv2014

Interval scheduling and colorful independent sets

Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer k, find a set of at least k pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an interval graph and a cluster graph), on which Independent Set remains NP-hard but admits constant-ratio approximations in polynomial time. We study the parameterized complexity of Independent Set on 2-union graphs and on subclasses like strip graphs. Our investigations significantly benefit from a new structural "compactness" parameter of interval graphs and novel problem formulations using vertex-colored interval graphs. Our main contributions are: 1. We show a complexity dichotomy: restricted to graph classes closed under induced subgraphs and disjoint unions, Independent Set is polynomial-time solvable if both input interval graphs are cluster graphs, and is NP-hard otherwise. 2. We chart the possibilities and limits of effective polynomial-time preprocessing (also known as kernelization). 3. We extend Halld\'orsson and Karlsson (2006)'s fixed-parameter algorithm for Independent Set on strip graphs parameterized by the structural parameter "maximum number of live jobs" to show that the problem (also known as Job Interval Selection) is fixed-parameter tractable with respect to the parameter k and generalize their algorithm from strip graphs to 2-union graphs. Preliminary experiments with random data indicate that Job Interval Selection with up to fifteen jobs and 5*10^5 intervals can be solved optimally in less than five minutes.

cs.DM↗