arXiv subjects

Matthias Bentert

Publications and source records attributed to Matthias Bentert.

At least 37 records · Page 2Linked to original sources

arXiv2025

Exponential-Time Approximation (Schemes) for Vertex-Ordering Problems

In this paper, we begin the exploration of vertex-ordering problems through the lens of exponential-time approximation algorithms. In particular, we ask the following question: Can we simultaneously beat the running times of the fastest known (exponential-time) exact algorithms and the best known approximation factors that can be achieved in polynomial time? Following the recent research initiated by Esmer et al. (ESA 2022, IPEC 2023, SODA 2024) on vertex-subset problems, and by Inamdar et al. (ITCS 2024) on graph-partitioning problems, we focus on vertex-ordering problems. In particular, we give positive results for Feedback Arc Set, Optimal Linear Arrangement, Cutwidth, and Pathwidth. Most of our algorithms build upon a novel ``balanced-cut'' approach, which is our main conceptual contribution. This allows us to solve various problems in very general settings allowing for directed and arc-weighted input graphs. Our main technical contribution is a (1+ε)-approximation for any ε > 0 for (weighted) Feedback Arc Set in O*((2-δ)^n) time, where δ > 0 is a constant only depending on ε.

cs.DS↗

arXiv2024

Packing Short Cycles

Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Björklund, Husfeldt, ICALP 2014; Mari, Mukherjee, Pilipczuk, and Sankowski, SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the following cycle packing problems: Min-Sum Cycle Packing and Shortest Cycle Packing. In Min-Sum Cycle Packing, we try to find, in a weighted undirected graph, $k$ vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed $k$, solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by $k$. The same results hold for the version of the problem where the task is to find $k$ edge-disjoint cycles. Our second main result concerns Shortest Cycle Packing, which is a special case of Min-Sum Cycle Packing that asks to find a packing of $k$ shortest cycles in a graph. We prove this problem to be fixed-parameter tractable (FPT) when parameterized by $k$ on weighted planar graphs. We also obtain a polynomial kernel for the edge-disjoint variant of the problem on planar graphs. Deciding whether Min-Sum Cycle Packing is FPT on planar graphs and whether Shortest Cycle Packing is FPT on general graphs remain challenging open questions.

cs.DS↗

arXiv2024

Leveraging Fixed-Parameter Tractability for Robot Inspection Planning

Autonomous robotic inspection, where a robot moves through its environment and inspects points of interest, has applications in industrial settings, structural health monitoring, and medicine. Planning the paths for a robot to safely and efficiently perform such an inspection is an extremely difficult algorithmic challenge. In this work we consider an abstraction of the inspection planning problem which we term Graph Inspection. We give two exact algorithms for this problem, using dynamic programming and integer linear programming. We analyze the performance of these methods, and present multiple approaches to achieve scalability. We demonstrate significant improvement both in path weight and inspection coverage over a state-of-the-art approach on two robotics tasks in simulation, a bridge inspection task by a UAV and a surgical inspection task using a medical robot.

cs.RO↗

arXiv2024

Correlation Clustering with Vertex Splitting

We explore Cluster Editing and its generalization Correlation Clustering with a new operation called permissive vertex splitting which addresses finding overlapping clusters in the face of uncertain information. We determine that both problems are NP-hard, yet they exhibit significant differences in parameterized complexity and approximability. For Cluster Editing with Permissive Vertex Splitting, we show a polynomial kernel when parameterized by the solution size and develop a polynomial-time algorithm with approximation factor 7. In the case of Correlation Clustering, we establish para-NP-hardness when parameterized by solution size and demonstrate that computing an $n^{1-ε}$-approximation is NP-hard for any constant $ε> 0$. Additionally, we extend the established link between Correlation Clustering and Multicut to the setting with permissive vertex splitting.

cs.DS↗

arXiv2024

The Parameterized Complexity Landscape of Two-Sets Cut-Uncut

In Two-Sets Cut-Uncut, we are given an undirected graph $G=(V,E)$ and two terminal sets $S$ and $T$. The task is to find a minimum cut $C$ in $G$ (if there is any) separating $S$ from $T$ under the following ``uncut'' condition. In the graph $(V,E \setminus C)$, the terminals in each terminal set remain in the same connected component. In spite of the superficial similarity to the classic problem Minimum $s$-$t$-Cut, Two-Sets Cut-Uncut is computationally challenging. In particular, even deciding whether such a cut of any size exists, is already NP-complete. We initiate a systematic study of Two-Sets Cut-Uncut within the context of parameterized complexity. By leveraging known relations between many well-studied graph parameters, we characterize the structural properties of input graphs that allow for polynomial kernels, fixed-parameter tractability (FPT), and slicewise polynomial algorithms (XP). Our main contribution is the near-complete establishment of the complexity of these algorithmic properties within the described hierarchy of graph parameters. On a technical level, our main results are fixed-parameter tractability for the (vertex-deletion) distance to cographs and an OR-cross composition excluding polynomial kernels for the vertex cover number of the input graph (under the standard complexity assumption NP is not contained in coNP/poly).

cs.DS↗

arXiv2024

The Structural Complexity Landscape of Finding Balance-Fair Shortest Paths

We study the parameterized complexity of finding shortest s-t-paths with an additional fairness requirement. The task is to compute a shortest path in a vertex-colored graph where each color appears (roughly) equally often in the solution. We provide a complete picture of the parameterized complexity landscape of the problem with respect to structural parameters by showing a tetrachotomy including polynomial kernels, fixed-parameter tractability, XP-time algorithms (and W[1]-hardness), and para-NP-hardness.

cs.DS↗

arXiv2024

Fully Polynomial-time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication

We study the computational complexity of several polynomial-time-solvable graph problems parameterized by vertex integrity, a measure of a graph's vulnerability to vertex removal in terms of connectivity. Vertex integrity is the smallest number $ι$ such that there is a set $S$ of $ι' \le ι$ vertices such that every connected component of $G-S$ contains at most $ι-ι'$ vertices. It is known that the vertex integrity lies between the well-studied parameters vertex cover number and tree-depth. Alon and Yuster [ESA 2007] designed algorithms for graphs with small vertex cover number using fast matrix multiplications. We demonstrate that fast matrix multiplication can also be effectively used when parameterizing by vertex integrity $ι$ by developing efficient algorithms for problems including an $O(ι^{ω-1}n)$-time algorithm for computing the girth of a graph, randomized $O(ι^{ω- 1}n)$-time algorithms for Maximum Matching and for finding any induced four-vertex subgraph except for a clique or an independent set, and an $O(ι^{(ω-1)/2}n^2) \subseteq O(ι^{0.687} n^2)$-time algorithm for All-Pairs Shortest Paths. These algorithms can be faster than previous algorithms parameterized by tree-depth, for which fast matrix multiplication is not known to be effective.

cs.DS↗

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

Cluster Editing with Vertex Splitting

Cluster Editing, also known as Correlation Clustering, is a well-studied graph modification problem. In this problem, one is given a graph and the task is to perform up to $k$ edge additions or deletions to transform it into a cluster graph, i.e., a graph consisting of a disjoint union of cliques. However, in real-world networks, clusters are often overlapping. For example in social networks, a person might belong to several communities - e.g. those corresponding to work, school, or neighborhood. Other strong motivations come from biological network analysis and from language networks. Trying to cluster words with similar usage in the latter can be confounded by homonyms, that is, words with multiple meanings like "bat." In this paper, we introduce a new variant of Cluster Editing whereby a vertex can be split into two or more vertices. First used in the context of graph drawing, this operation allows a vertex $v$ to be replaced by two vertices whose combined neighborhood is the neighborhood of $v$ (and thus $v$ can belong to more than one cluster). We call the new problem Cluster Editing with Vertex Splitting and we initiate the study of it. We show that it is NP-complete and fixed-parameter tractable when parameterized by the total number $k$ of allowed vertex-splitting and edge-editing operations. In particular, we obtain an $O(2^{9k log k} + n + m)$-time algorithm and a $6k$-vertex kernel.

cs.CC↗

arXiv2023

Two-sets cut-uncut on planar graphs

We study the following Two-Sets Cut-Uncut problem on planar graphs. Therein, one is given an undirected planar graph $G$ and two sets of vertices $S$ and $T$. The question is, what is the minimum number of edges to remove from $G$, such that we separate all of $S$ from all of $T$, while maintaining that every vertex in $S$, and respectively in $T$, stays in the same connected component. We show that this problem can be solved in time $2^{|S|+|T|} n^{O(1)}$ with a one-sided error randomized algorithm. Our algorithm implies a polynomial-time algorithm for the network diversion problem on planar graphs, which resolves an open question from the literature. More generally, we show that Two-Sets Cut-Uncut remains fixed-parameter tractable even when parameterized by the number $r$ of faces in the plane graph covering the terminals $S \cup T$, by providing an algorithm of running time $4^{r + O(\sqrt r)} n^{O(1)}$.

cs.DS↗

arXiv2023

Fair Short Paths in Vertex-Colored Graphs

The computation of short paths in graphs with arc lengths is a pillar of graph algorithmics and network science. In a more diverse world, however, not every short path is equally valuable. For the setting where each vertex is assigned to a group (color), we provide a framework to model multiple natural fairness aspects. We seek to find short paths in which the number of occurrences of each color is within some given lower and upper bounds. Among other results, we prove the introduced problems to be computationally intractable (NP-hard and parameterized hard with respect to the number of colors) even in very restricted settings (such as each color should appear with exactly the same frequency), while also presenting an encouraging algorithmic result ("fixed-parameter tractability") related to the length of the sought solution path for the general problem.

cs.DS↗

arXiv2022

Polynomial-Time Data Reduction for Weighted Problems Beyond Additive Goal Functions

Dealing with NP-hard problems, kernelization is a fundamental notion for polynomial-time data reduction with performance guarantees: in polynomial time, a problem instance is reduced to an equivalent instance with size upper-bounded by a function of a parameter chosen in advance. Kernelization for weighted problems particularly requires to also shrink weights. Marx and Végh [ACM Trans. Algorithms 2015] and Etscheid et al. [J. Comput. Syst. Sci. 2017] used a technique of Frank and Tardos [Combinatorica 1987] to obtain polynomial-size kernels for weighted problems, mostly with additive goal functions. We characterize the function types that the technique is applicable to, which turns out to contain many non-additive functions. Using this insight, we systematically obtain kernelization results for natural problems in graph partitioning, network design, facility location, scheduling, vehicle routing, and computational social choice, thereby improving and generalizing results from the literature.

cs.DS↗

arXiv2022

Who won? Winner Determination and Robustness in Liquid Democracy

Liquid democracy is a decision-making paradigm in which each agent can either vote directly for some alternative or (transitively) delegate its vote to another agent. To mitigate the issue of delegation cycles or the concentration of power, delegating agents might be allowed to specify multiple delegation options. Then, a (cycle-free) delegation is selected in which each delegating agent has exactly one representative. We study the winner determination problem for this setting, i.e., whether we can select a delegation such that a given alternative wins (or does not win). Moreover, we study the robustness of winning alternatives in two ways: First, we consider whether we can make a limited number of changes to the preferences cast by the agents such that a given alternative becomes a winner in one/in all delegations, and second, whether we can make a limited number of changes to a selected delegation to make a given alternative a winner.

cs.GT↗

arXiv2022

Stable Matching with Multilayer Approval Preferences: Approvals can be Harder than Strict Preferences

We study stable matching problems where agents have multilayer preferences: There are $\ell$ layers each consisting of one preference relation for each agent. Recently, Chen et al. [EC '18] studied such problems with strict preferences, establishing four multilayer adaptions of classical notions of stability. We follow up on their work by analyzing the computational complexity of stable matching problems with multilayer approval preferences. We consider eleven stability notions derived from three well-established stability notions for stable matchings with ties and the four adaptions proposed by Chen et al. For each stability notion, we show that the problem of finding a stable matching is either polynomial-time solvable or NP-hard. Furthermore, we examine the influence of the number of layers and the desired "degree of stability" on the problems' complexity. Somewhat surprisingly, we discover that assuming approval preferences instead of strict preferences does not considerably simplify the situation (and sometimes even makes polynomial-time solvable problems NP-hard).

cs.GT↗

arXiv2022

Using a geometric lens to find k disjoint shortest paths

Given an undirected $n$-vertex graph and $k$ pairs of terminal vertices $(s_1,t_1), \ldots, (s_k,t_k)$, the $k$-Disjoint Shortest Paths ($k$-DSP)-problem asks whether there are $k$ pairwise vertex-disjoint paths $P_1,\ldots, P_k$ such that $P_i$ is a shortest $s_i$-$t_i$-path for each $i \in [k]$. Recently, Lochet [SODA 2021] provided an algorithm that solves $k$-DSP in $n^{O(k^{5^k})}$ time, answering a 20-year old question about the computational complexity of $k$-DSP for constant $k$. On the one hand, we present an improved $n^{O(k!k)}$-time algorithm based on a novel geometric view on this problem. For the special case $k=2$ on $m$-edge graphs, we show that the running time can be further reduced to $O(nm)$ by small modifications of the algorithm and a refined analysis. On the other hand, we show that $k$-DSP is W[1]-hard with respect to $k$, showing that the dependency of the degree of the polynomial running time on the parameter $k$ is presumably unavoidable.

math.CO↗

arXiv2021

A Multivariate Complexity Analysis of the Material Consumption Scheduling Problem

The NP-hard MATERIAL CONSUMPTION SCHEDULING Problem and closely related problems have been thoroughly studied since the 1980's. Roughly speaking, the problem deals with minimizing the makespan when scheduling jobs that consume non-renewable resources. We focus on the single-machine case without preemption: from time to time, the resources of the machine are (partially) replenished, thus allowing for meeting a necessary pre-condition for processing further jobs, each of which having individual resource demands. We initiate a systematic exploration of the parameterized (exact) complexity landscape of the problem, providing parameterized tractability as well as intractability results. Doing so, we mainly investigate how parameters related to the resource supplies influence the computational solvability. Thereby, we get a deepened understanding of the algorithmic complexity of this fundamental scheduling problem.

cs.GT↗

arXiv2021

The Complexity of Gerrymandering Over Graphs: Paths and Trees

Roughly speaking, gerrymandering is the systematic manipulation of the boundaries of electoral districts to make a specific (political) party win as many districts as possible. While typically studied from a geographical point of view, addressing social network structures, the investigation of gerrymandering over graphs was recently initiated by Cohen-Zemach et al. [AAMAS 2018]. Settling three open questions of Ito et al. [AAMAS 2019], we classify the computational complexity of the NP-hard problem Gerrymandering over Graphs when restricted to paths and trees. Our results, which are mostly of negative nature (that is, worst-case hardness), in particular yield two complexity dichotomies for trees. For instance, the problem is polynomial-time solvable for two parties but becomes weakly NP-hard for three. Moreover, we show that the problem remains NP-hard even when the input graph is a path.

cs.DS↗

arXiv2020

Parameterized Complexity of Diameter

Diameter -- the task of computing the length of a longest shortest path -- is a fundamental graph problem. Assuming the Strong Exponential Time Hypothesis, there is no $O(n^{1.99})$-time algorithm even in sparse graphs [Roditty and Williams, 2013]. To circumvent this lower bound we aim for algorithms with running time $f(k)(n+m)$ where $k$ is a parameter and $f$ is a function as small as possible. We investigate which parameters allow for such running times. To this end, we systematically explore a hierarchy of structural graph parameters.

cs.DS↗