# Till Fluschnik's research while affiliated with Technische Universität Clausthal and other places

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## Publications (72)

We study the computational complexity of identifying a small agreeable subset of items. A subset of items is agreeable if every agent does not prefer its complement set. We study settings in which agents either can assign arbitrary utilities to the items; can approve or disapprove the items; or can rank the items (in which case we consider Borda ut...

We study the election of sequences of committees, where in each of tau levels (e.g. modeling points in time) a committee consisting of k candidates from a common set of m candidates is selected. For each level, each of n agents (voters) may nominate one candidate whose selection would satisfy her. We are interested in committees which are good with...

We study the election of sequences of committees, where in each of $\tau$ levels (e.g. modeling points in time) a committee consisting of $k$ candidates from a common set of $m$ candidates is selected. For each level, each of $n$ agents (voters) may nominate one candidate whose selection would satisfy her. We are interested in committees which are...

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 shr...

Addressing a quest by Gupta et al. (in: Proceedings of the 41st international colloquium on automata, languages, and programming (ICALP 2014), vol 8572 of LNCS. Springer, pp 563–575, 2014), we provide a first, comprehensive study of finding a short s – t path in the multistage graph model, referred to as the Multistage s – t Path problem. Herein, g...

Choosing the placement of wildlife crossings (i.e., green bridges) to reconnect animal species' fragmented habitats is among the 17 goals towards sustainable development by the UN. We consider the following established model: Given a graph whose vertices represent the fragmented habitat areas and whose weighted edges represent possible green bridge...

Electing a single committee of a small size is a classical and well-understood voting situation. Being interested in a sequence of committees, we introduce two time-dependent multistage models based on simple scoring-based voting. Therein, we are given a sequence of voting profiles (stages) over the same set of agents and candidates, and our task i...

The NP-complete Vertex Cover problem asks to cover all edges of a graph by a small (given) number of vertices. It is among the most prominent graph-algorithmic problems. Following a recent trend in studying temporal graphs (a sequence of graphs, so-called layers, over the same vertex set but, over time, changing edge sets), we initiate the study of...

Choosing the placement of wildlife crossings (i.e., green bridges) to reconnect animal species' fragmented habitats is among the 17 goals towards sustainable development by the UN. We consider the following established model: Given a graph whose vertices represent the fragmented habitat areas and whose weighted edges represent possible green bridge...

We consider the algorithmic complexity of recognizing bipartite temporal graphs. Rather than defining these graphs solely by their underlying graph or individual layers, we define a bipartite temporal graph as one in which every layer can be 2-colored in a way that results in few changes between any two consecutive layers. This approach follows the...

A polynomial Turing kernel for some parameterized problem $P$ is a polynomial-time algorithm that solves $P$ using queries to an oracle of $P$ whose sizes are upper-bounded by some polynomial in the parameter. Here the term "polynomial" refers to the bound on the query sizes, as the running time of any kernel is required to be polynomial. One of th...

We study the computational complexity of Feedback Vertex Set on subclasses of Hamiltonian graphs. In particular, we consider Hamiltonian graphs that are regular or are planar and regular. Moreover, we study the less known class of p-Hamiltonian-ordered graphs, which are graphs that admit for any p-tuple of vertices a Hamiltonian cycle visiting them...

Proximity graphs have been studied for several decades, motivated by applications in computational geometry, geography, data mining, and many other fields. However, the computational complexity of classic graph problems on proximity graphs mostly remained open. We now study 3-Colorability, Dominating Set, Feedback Vertex Set, Hamiltonian Cycle, and...

We study the problem of placing wildlife crossings, such as green bridges, over human-made obstacles to challenge habitat fragmentation. The main task herein is, given a graph describing habitats or routes of wildlife animals and possibilities of building green bridges, to find a low-cost placement of green bridges that connects the habitats. We de...

We study q-SAT in the multistage model, focusing on the linear-time solvable 2-SAT. Herein, given a sequence of q-CNF formulas and a non-negative integer d, the question is whether there is a sequence of satisfying truth assignments such that for every two consecutive truth assignments, the number of variables whose values changed is at most d. We...

We prove that 3-Coloring remains NP-hard on 4- and 5-regular planar Hamiltonian graphs, strengthening the results of Dailey [Disc. Math.'80] and Fleischner and Sabidussi [J. Graph. Theor.'02]. Moreover, we prove that 3-Coloring remains NP-hard on $p$-regular Hamiltonian graphs for every $p\geq 6$ and $p$-ordered regular Hamiltonian graphs for every...

We study the computational complexity of Feedback Vertex Set on subclasses of Hamiltonian graphs. In particular, we consider Hamiltonian graphs that are regular or are planar and regular. Moreover, we study the less known class of $p$-Hamiltonian-ordered graphs, which are graphs that admit for any $p$-tuple of vertices a Hamiltonian cycle visiting...

We study the problem of placing wildlife crossings, such as green bridges, over human-made obstacles to challenge habitat fragmentation. The main task herein is, given a graph describing habitats or routes of wildlife animals and possibilities of building green bridges, to find a low-cost placement of green bridges that connects the habitats. We de...

Kernelization is the fundamental notion for polynomial-time prepocessing with performance guarantees in parameterized algorithmics. When preprocessing weighted problems, the need of shrinking weights might arise. Marx and V\'egh [ACM Trans. Algorithms 2015] and Etscheid et al. [J. Comput. Syst. Sci. 2017] used a technique due to Frank and Tardos [C...

Presentation of article Networks 76(4):485–508, 2020

We study $q$-SAT in the multistage model, focusing on the linear-time solvable 2-SAT. Herein, given a sequence of $q$-CNF fomulas and a non-negative integer $d$, the question is whether there is a sequence of satisfying truth assignments such that for every two consecutive truth assignments, the number of variables whose values changed is at most $...

Given an undirected graph with edge weights and a subset R of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. We prove that RPP is WK[1]-complete parameterized by the number and weight d of edges traversed additionally to the required ones. Thus RPP instances cannot be polynomia...

Given an undirected graph with edge weights and a subset R of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. We prove that RPP is WK[1]-complete parameterized by the number and cost d of edges traversed additionally to the required ones. Thus, in particular, RPP instances canno...

Electing a single committee of a small size is a classical and well-understood voting situation. Being interested in a sequence of committees, we introduce and study two time-dependent multistage models based on simple Plurality voting. Therein, we are given a sequence of voting profiles (stages) over the same set of agents and candidates, and our...

Treewidth is arguably the most important structural graph parameter leading to algorithmically beneficial graph decompositions. Triggered by a strongly growing interest in temporal networks (graphs where edge sets change over time), we discuss fresh algorithmic views on temporal tree decompositions and temporal treewidth. We review and explain some...

Treewidth is arguably the most important structural graph parameter leading to algorithmically beneficial graph decompositions. Triggered by a strongly growing interest in temporal networks (graphs where edge sets change over time), we discuss fresh algorithmic views on temporal tree decompositions and temporal treewidth. We review and explain some...

Finding paths in graphs is a fundamental graph-theoretic task. In this work, we are concerned with finding a path with some constraints on its length and the number of vertices neighboring the path, that is, being outside of and incident with the path. Herein, we consider short and long paths on the one side, and small and large neighborhoods on th...

Addressing a quest by Gupta et al. [ICALP'14], we provide a first, comprehensive study of finding a short s-t path in the multistage graph model, referred to as the Multistage s-t Path problem. Herein, given a sequence of graphs over the same vertex set but changing edge sets, the task is to find short s-t paths in each graph ("snapshot") such that...

Given a graph G = (V, E), two vertices s, t ∈ V, and two integers k, ℓ, the Short Secluded Path problem is to find a simple s‐t‐path with at most k vertices and ℓ neighbors. We study the parameterized complexity of the problem with respect to four structural graph parameters: the vertex cover number, treewidth, feedback vertex number, and feedback...

Temporal graphs have time-stamped edges. Building on previous work, we study the problem of finding a small vertex set (the separator) whose removal destroys all temporal paths between two designated terminal vertices. Herein, we consider two models of temporal paths: those that pass through arbitrarily many edges per time step (non-strict) and tho...

Given a graph \(G=(V,E)\) with edge weights and a subset \(R\subseteq E\) of required edges, the NP-hard Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. The number b of vertices incident to an odd number of edges of R and the number c of connected components formed by the edges in R are both b...

Given a graph G=(V,E), two vertices s,t∈V, and two integers k,ℓ, the Short Secluded Path problem is to find a simple s-t-path with at most k vertices and ℓ neighbors. We study the parameterized complexity of the problem with respect to four structural graph parameters: the vertex cover number, treewidth, feedback vertex number, and feedback edge nu...

Covering all edges of a graph by a minimum number of vertices, this is the NP-hard Vertex Cover problem, is among the most fundamental algorithmic tasks. Following a recent trend in studying dynamic and temporal graphs, we initiate the study of Multistage Vertex Cover. Herein, having a series of graphs with same vertex set but over time changing ed...

We study the computational complexity of routing multiple objects through a network while avoiding collision: Given a graph G with two distinct terminals and two positive integers p,k, the question is whether one can connect the terminals by at least p routes (walks, trails, paths; the latter two without repeated edges or vertices, respectively) su...

Hyperbolicity is a distance-based measure of how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms used in practice for computing the hyperbolicity number of an n-vertex graph have running time \(O(n^4)...

The task of listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) a...

Averaging time series under dynamic time warping is an important tool for improving nearest-neighbor classifiers and formulating centroid-based clustering. The most promising approach poses time series averaging as the problem of minimizing a Fréchet function. Minimizing the Fréchet function is NP-hard and so far solved by several heuristics and in...

This work studies the parameterized complexity of finding secluded solutions to classical combinatorial optimization problems on graphs such as finding minimum s-t separators, feedback vertex sets, dominating sets, maximum independent sets, and vertex deletion problems for hereditary graph properties: Herein, one searches not only to minimize or ma...

We investigate the computational complexity of separating two distinct vertices s and z by vertex deletion in a temporal graph. In a temporal graph, the vertex set is fixed but the edges have (discrete) time labels. Since the corresponding Temporal (s, z)-Separation problem is NP-hard, it is natural to investigate whether relevant special cases exi...

We study a problem that models safely routing a convoy through a transportation network, where any vertex adjacent to the travel path of the convoy requires additional precaution: Given a graph G = (V, E), two vertices s, t ∈ V , and two integers k, , we search for a simple s-t-path with at most k vertices and at most neighbors. We study the proble...

Finding paths in graphs is a fundamental graph-theoretic task. In this work, we study the task of finding a path with some constraints on its length and the number of vertices neighboring the path, that is, being outside of and incident with the path. Herein, we consider short and long path on the one side, and small and large neighborhoods on the...

Kernelization is an important tool in parameterized algorithmics. Given an input instance accompanied by a parameter, the goal is to compute in polynomial time an equivalent instance of the same problem such that the size of the reduced instance only depends on the parameter and not on the size of the original instance. In this paper, we provide a...

Kernelization—a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems—plays a central role in parameterized complexity and has triggered an extensive line of research. In this paper we consider a restricted yet natural variant of kernelization, namely strict kernelization, where one is not allowed to incr...

We study the NP-hard Shortest Path Most Vital Edges problem arising in the context of analyzing network robustness. For an undirected graph with positive integer edge lengths and two designated vertices $s$ and $t$, the goal is to delete as few edges as possible in order to increase the length of the (new) shortest $st$-path as much as possible. Th...

We investigate the computational complexity of separating two distinct vertices s and z by vertex deletion in a temporal graph. In a temporal graph, the vertex set is fixed but the edges have (discrete) time labels. Since the corresponding Temporal (s, z)-Separation problem is NP-hard, it is natural to investigate whether relevant special cases exi...

Dynamic time warping constitutes a major tool for analyzing time series. In particular, computing a mean series of a given sample of series in dynamic time warping spaces (by minimizing the Fr\'echet function) is a challenging computational problem, so far solved by several heuristic, inexact strategies. We spot several inaccuracies in the literatu...

The composition technique is a popular method for excluding polynomial-size problem kernels for NP-hard parameterized problems. We present a new technique exploiting triangle-based fractal structures for extending the range of applicability of compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of pr...

We study the following multiagent variant of the knapsack problem. We are given a set of items, a set of voters, and a value of the budget; each item is endowed with a cost and each voter assigns to each item a certain value. The goal is to select a subset of items with the total cost not exceeding the budget, in a way that is consistent with the v...

Vertex separators, that is, vertex sets whose deletion disconnects two distinguished vertices in a graph, play a pivotal role in algorithmic graph theory. For instance, the concept of tree decompositions of graphs is tightly connected to the separator concept. For many realistic models of the real world, however, it is necessary to consider graphs...

We study the \({\mathsf {NP}}\)-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route p paths from a start vertex to a target vertex in a given graph while using at most k edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or l...

Dynamic time warping constitutes a major tool for analyzing time series. In particular, computing a mean series of a given sample of series in dynamic time warping spaces (by minimizing the Fr\'echet function) is a challenging computational problem, so far solved by several heuristic, inexact strategies. We spot several inaccuracies in the literatu...

Kernelization is an important tool in parameterized algorithmics. The goal is to reduce the input instance of a parameterized problem in polynomial time to an equivalent instance of the same problem such that the size of the reduced instance only depends on the parameter and not on the size of the original instance. In this paper, we provide a firs...

Hyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known practical algorithms for computing the hyperbolicity number of a n-vertex graph have running time \(O(n...

We study the computational complexity of routing multiple objects through a network in such a way that only few collisions occur: Given a graph $G$ with two distinct terminal vertices and two positive integers $p$ and $k$, the question is whether one can connect the terminals by at least $p$ routes (e.g. paths) such that at most $k$ edges are time-...

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded grids in linear time when both dimensions are either small or large compared to the numbe...

Hyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms for computing the hyperbolicity number of a graph (the smaller, the more tree-like) have run...

Listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) as well pract...

Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the "exposure" of the solution to the...

Kernelization -- the mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems -- plays a central role in parameterized complexity and has triggered an extensive line of research. This is in part due to a lower bounds framework that allows to exclude polynomial-size kernels under the assumption that NP is not...

Human development has far-reaching impacts on the surface of the globe. The transformation of natural land cover occurs in different forms, and urban growth is one of the most eminent transformative processes. We analyze global land cover data and extract cities as defined by maximally connected urban clusters. The analysis of the city size distrib...

Cities play a vital role in the global climate change mitigation agenda. City population density is one of the key factors that influence urban energy consumption and the subsequent GHG emissions. However, previous research on the relationship between population density and GHG emissions led to contradictory results due to urban/rural definition co...

We study the NP-complete Minimum Shared Edges (MSE) problem. Given an undirected graph, a source and a sink vertex, and two integers p and k, the question is whether there are p paths in the graph connecting the source with the sink and sharing at most k edges. Herein, an edge is shared if it appears in at least two paths. We show that MSE is W[1]-...

We study the Minimum Shared Edges problem introduced by Omran et al. [Journal of Combinatorial Optimization, 2015] on planar graphs: Planar MSE asks, given a planar graph G = (V,E), two distinct vertices s,t in V , and two integers p, k, whether there are p s-t paths in G that share at most k edges, where an edges is called shared if it appears in...

Bodlaender et al.'s [SIDMA 2014] cross-composition technique is a popular
method for excluding polynomial-size problem kernels for NP-hard parameterized
problems. We present a new technique exploiting triangle-based fractal
structures for extending the range of applicability of cross-compositions. Our
technique makes it possible to prove new no-pol...

Human development has far-reaching impacts on the surface of the globe.
The transformation of natural land cover has different forms and besides
agricultural practices, urban growth is an eminent transformative
process. We analyze global land cover data and extract cities as defined
by maximally connected urban clusters. The analysis of city size
d...

## Citations

... We will use the following lemma to shrink edge weights so that their encoding length will be polynomial in the number of vertices and edges of the graph. It is a generalization of an idea implicitly used for weight reduction in a proof of Lokshtanov et al. [47,Theorem 4.2] and shrinks weights faster and more significantly than a theorem of Frank and Tardos [31] that is frequently used in the exact kernelization of weighted problems [3,6,25,48]. We first state the lemma, and thereafter intuitively describe its application to RPP. ...

... This introduces non-trivial dependencies between the time warping and the vertex mapping which render the problem computationally hard. Indeed, this is not a singular case for temporal graph problems where for many cases the temporal counterparts of problems solvable in polynomial time turn NP-hard; examples include the computation of matchings in graphs (Heeger et al. 2019;Baste et al. 2020;Mertzios et al. 2020), short path computations (Casteigts et al. 2019;Fluschnik et al. 2020), or the computation of separators . ...

... However, in multi-winner voting, a candidate cannot be usually elected multiple times, as in our setting (but see an exception in [9]). Notably, Bredereck et al. [8] study a sequence of multi-winner elections, in which the difference between the winners in consecutive rounds is upper-bounded. The framework of participatory budgeting [3,27], which generalizes multi-winner voting, utilizes voting systems for deciding on the funding of public projects. ...

... In another approach, MVCP is presented in an approach that solves the used graph by separating it into subgraphs [41]. MVCP is solved in the graph divided into layers with the multistage vertex-cover approach [42]. In the approach, the solution is reasonable under some constraints unless there are too many (too different) mismatched nodes in both layers. ...

... Related Work. In our previous work [4], we proved that Feedback Vertex Set remains NP-complete on 4-and 5-regular planar Hamiltonian graphs, p-regular Hamiltonian graphs for every p ∈ N ≥6 , and p-Hamiltonian-ordered graphs for every p ∈ N ≥3 . Several classic graph problems are studied on planar regular graphs. ...

... We follow a model recently introduced by Fluschnik and Kellerhals [6]. Herein, the modeled graph can be understood as path-based graph [22,8]: a vertex corresponds to a part fragmented by human-made infrastructures subsuming habitat patches of diverse animal habitats, and any two vertices are connected by an edge if the corresponding patches can be connected by a green bridge. ...

... , I τ ) of instances of some problem P as input, and it asks for a "robust" sequence of solutions to the instances in the sense that any two consecutive solutions are similar. Several classical problems have been studied in the multistage model, both from an approximate [1][2][3][4] and from a parameterized [8,19,20,23,31,34] algorithmics point of view. While E E-MstP and V V-MstP adhere to the original multistage model, our two problems E∩E-MstP and V∩V-MstP can be seen as a novel and natural variation of the multistage model by replacing the goal of consecutive similarity with consecutive dissimilarity. ...

... More generally, to encompass the computation of approximate solutions with a loss of factor α ≥ 1, given a β-approximate solution to the instance of Q, for any β ≥ 1, lift must output an α · β-approximate solution to the instance of P . Since its introduction, this notion of compression/kernelization (termed lossy compression/kernelization) has already found a wide range of applications; see, e.g., [40,28,21,36,35,1,47,20] for just a few illustrative examples. ...

Reference: Kernelization of Counting Problems

... The standard approach, also used in kernelization of weighted problems [6,8,9,16,25,33,37] is to use the above proposition which "kernelizes" a linear objective function if the dimension is bounded by a parameter. ...

... This motivates the development of parameters that take into account the temporal structure of the input, as well as the structure of the underlying graph. Some measures of this kind (such as temporal variants of feedback vertex number [10] and treewidth [21]) have already been studied. Unfortunately we find that these parameters are of no use to us since the problems we consider here remain NP-complete even when these measures are bounded by constants on the underlying static graph. ...

Reference: Edge Exploration of Temporal Graphs