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Anton Herrmann

Publications and source records attributed to Anton Herrmann.

6 recordsLinked to original sources
arXiv2026

Parameterized Complexity of Temporal Agony

Real-world networks are often organized in several layers forming a hierarchy which determines the interaction between the individual components. In order to discover such hierarchies in temporal networks, Tatti [ECML PKDD 2018] introduced the temporal agony problem Seg-Agony. Here, the goal is to assign each vertex a certain rank (from 1 to $k$) such that arcs only point from lower ranks to higher ranks. Backward arcs are penalized depending on the difference between the corresponding ranks. Since arcs may change over time, each vertex is allowed to change its rank $\ell\ge 1$ times in order to minimize the overall penalty $\alpha$ (called temporal agony). We study the parameterized complexity of Seg-Agony with a special focus on the number $k$ of possible ranks for which we identify the precise complexity border. We show that the problem is polynomial-time solvable for $k=2$, NP-hard for $k=3$ and $\ell=1$ but polynomial-time solvable for constant $\alpha$, and NP-hard for $k=4$ and $\ell=1$ even for $\alpha=0$. We further show a polynomial-time algorithm for a constant number $n$ of vertices and fixed-parameter tractability for the combined parameter $n+\ell$.

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arXiv2026

On the Parameterized Complexity of Bounded-Density Vertex Deletion

We explore the parameterized complexity of Bounded Density Vertex Deletion (BDVD): given a graph $G$, an integer budget $k$, and a target density $\tau_\rho$, the task is to determine whether the density (i.e. number of edges divided by number of vertices) of the densest subgraph of $G$ can be reduced to at most $\tau_\rho$ by deleting at most $k$ vertices. Our primary focus is on structural graph parameters related to treewidth, as the parameterized complexity of BDVD with respect to treewidth was left as open question by Bazgan et al. [JCSS, 2025]. We resolve this question by showing W[1]-hardness with respect to various parameters, including treedepth and feedback vertex number. These results imply W[1]-hardness with respect to treewidth. We obtain positive results for parameters larger than treedepth and feedback vertex number, namely we show BDVD is in FPT parameterized by the max leaf number or vertex integrity. Under the assumption that the target density $\tau_\rho$ is a fixed constant the parameterized complexity landscape of BDVD changes drastically, allowing a fixed-parameter tractable algorithm even for parameters smaller than treewidth, namely cliquewidth. Altogether, our results provide a refined complexity landscape for Bounded Density Vertex Deletion, sharply distinguishing between tractable and intractable parameter regimes under structural parameterizations.

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arXiv2026

Parameterized Algorithms for Computing MAD Trees

We consider the well-studied problem of finding a spanning tree with minimum average distance between vertex pairs (called a MAD tree). This is a classic network design problem which is known to be NP-hard. While approximation algorithms and polynomial-time algorithms for some graph classes are known, the parameterized complexity of the problem has not been investigated so far. We start a parameterized complexity analysis with the goal of determining the border of algorithmic tractability for the MAD tree problem. To this end, we provide a linear-time algorithm for graphs of constant modular width and a polynomial-time algorithm for graphs of bounded treewidth; the degree of the polynomial depends on the treewidth. That is, the problem is in FPT with respect to modular width and in XP with respect to treewidth. Moreover, we show it is in FPT when parameterized by vertex integrity or by an above-guarantee parameter. We complement these algorithms with NP-hardness on split graphs.

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arXiv2026

Density Matters: A Complexity Dichotomy of Deleting Edges to Bound Subgraph Density

We study $\tau$-Bounded-Density Edge Deletion ($\tau$-BDED), where given an undirected graph $G$, the task is to remove as few edges as possible to obtain a graph $G'$ where no subgraph of $G'$ has density more than $\tau$. The density of a (sub)graph is the number of edges divided by the number of vertices. This problem was recently introduced and shown to be NP-hard for $\tau \in \{2/3, 3/4, 1 + 1/25\}$, but polynomial-time solvable for $\tau \in \{0,1/2,1\}$ [Bazgan et al., JCSS 2025]. We provide a complete dichotomy with respect to the target density $\tau$: 1. If $2\tau \in \mathbb{N}$ (half-integral target density) or $\tau < 2/3$, then $\tau$-BDED is polynomial-time solvable. 2. Otherwise, $\tau$-BDED is NP-hard. We complement the NP-hardness with fixed-parameter tractability with respect to the treewidth of $G$. Moreover, for integral target density $\tau \in \mathbb{N}$, we show $\tau$-BDED to be solvable in randomized $O(m^{1 + o(1)})$ time. Our algorithmic results are based on a reduction to a new general flow problem on restricted networks that, depending on $\tau$, can be solved via Maximum s-t-Flow or General Factors. We believe this connection between these variants of flow and matching to be of independent interest.

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arXiv2025

Timeline Problems in Temporal Graphs: Vertex Cover vs. Dominating Set

A temporal graph is a finite sequence of graphs, called snapshots, over the same vertex set. Many temporal graph problems turn out to be much more difficult than their static counterparts. One such problem is \textsc{Timeline Vertex Cover} (also known as \textsc{MinTimeline$_\infty$}), a temporal analogue to the classical \textsc{Vertex Cover} problem. In this problem, one is given a temporal graph $\mathcal{G}$ and two integers $k$ and $\ell$, and the goal is to cover each edge of each snapshot by selecting for each vertex at most $k$ activity intervals of length at most $\ell$ each. Here, an edge $uv$ in the $i$th snapshot is covered, if an activity interval of $u$ or $v$ is active at time $i$. In this work, we continue the algorithmic study of \textsc{Timeline Vertex Cover} and introduce the \textsc{Timeline Dominating Set} problem where we want to dominate all vertices in each snapshot by the selected activity intervals. We analyze both problems from a classical and parameterized point of view and also consider partial problem versions, where the goal is to cover (dominate) at least $t$ edges (vertices) of the snapshots. With respect to the parameterized complexity, we consider the temporal graph parameters vertex-interval-membership-width $(vimw)$ and interval-membership-width $(imw)$. We show that all considered problems admit FPT-algorithms when parameterized by $vimw + k+\ell$. This provides a smaller parameter combination than the ones used for previously known FPT-algorithms for \textsc{Timeline Vertex Cover}. Surprisingly, for $imw+ k+\ell$, \textsc{Timeline Dominating Set} turns out to be easier than \textsc{Timeline Vertex Cover}, by also admitting an FPT-algorithm, whereas the vertex cover version is NP-hard even if $imw+\, k+\ell$ is constant. We also consider parameterization by combinations of $n$, the vertex set size, with $k$ or $\ell$ and parameterization by $t$.

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