Till Fluschnik’s research while affiliated with Technische Universität Berlin and other places

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Publications (80)


A diagram of interconnections between the problems (for the definition of Connect GBP see Problem 1). An edge from problem A to problem B means that any solution to A is also a solution to B. Problems with d omitted from the problem name require that there is a solution for some value of d
Illustration to Construction 1 for 1-Reach GBP on series-parallel (and thus planar) graphs. In this example, there are e.g. Fp⊇{1,i,j}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_p\supseteq \{1,i,j\}$$\end{document} and Fq⊇{i,j,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_q\supseteq \{i,j,n\}$$\end{document}. In case of a yes-instance, the red-colored edges are in every solution (Observation 3).
Illustration to Constructions 2 and 3. Part (a) shows an exemplary directed graph which is a yes-instance for DHP. Applying Construction 2 on (a) yields (b). Applying Construction 3 on (b) yields the instance whose graph is depicted in (c) and two habitats of which are depicted in (d) and (e). Vertices marked yellow in (d) are contained in the habitat Xout\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_\textrm{out}$$\end{document}. Vertices marked red in (e) are contained in the habitat Yout\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_\textrm{out}$$\end{document}. The graph induced by Yout\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_\textrm{out}$$\end{document} contains the red edges
Illustration for 2-Reach GBP with (a) r=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=2$$\end{document} and Δ=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta =4$$\end{document} (k′=m+(n-1)+k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k'=m+(n-1)+k$$\end{document}) and (b) r=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=1$$\end{document} (k′=m+k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k'=m+k$$\end{document})
Illustration for the construction in the proof of Proposition 7 for 2-Reach GBP with r=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=1$$\end{document}. In this example, U={u1,⋯,un}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U=\{u_1,\dots ,u_n\}$$\end{document} and we have {u1,ui,uj,un}=F∈F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{u_1,u_i,u_j,u_n\}= F\in \mathscr {F}$$\end{document}

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Placing Green Bridges Optimally, with a Multivariate Analysis
  • Article
  • Full-text available

April 2024

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21 Reads

Theory of Computing Systems

Till Fluschnik

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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 develop three problem models for this task and study them from a computational complexity and parameterized algorithmics perspective.

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Locally Rainbow Paths

March 2024

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15 Reads

Proceedings of the AAAI Conference on Artificial Intelligence

We introduce the algorithmic problem of finding a locally rainbow path of length l connecting two distinguished vertices s and t in a vertex-colored directed graph. Herein, a path is locally rainbow if between any two visits of equally colored vertices, the path traverses consecutively at leaset r differently colored vertices. This problem generalizes the well-known problem of finding a rainbow path. It finds natural applications whenever there are different types of resources that must be protected from overuse, such as crop sequence optimization or production process scheduling. We show that the problem is computationally intractable even if r=2 or if one looks for a locally rainbow among the shortest paths. On the positive side, if one looks for a path that takes only a short detour (i.e., it is slightly longer than the shortest path) and if r is small, the problem can be solved efficiently. Indeed, the running time of the respective algorithm is near-optimal unless the ETH fails.


Efficiently Computing Smallest Agreeable Sets

September 2023

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19 Reads

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 utilities). We prove that deciding whether an agreeable set exists is NP-hard for all variants; and we perform a parameterized analysis regarding the following natural parameters: the number of agents, the number of items, and the upper bound on the size of the agreeable set in question.


Algorithmics of Egalitarian versus Equitable Sequences of Committees

August 2023

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7 Reads

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3 Citations

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 respect to the satisfaction per day and per agent. More precisely, we look for egalitarian or equitable committee sequences. While both guarantee that at least x agents per day are satisfied, egalitarian committee sequences ensure that each agent is satisfied in at least y levels while equitable committee sequences ensure that each agent is satisfied in exactly y levels. We analyze the parameterized complexity of finding such committees for the parameters n, m, k, tau, x, and y, as well as combinations thereof.


Algorithmics of Egalitarian versus Equitable Sequences of Committees

June 2023

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3 Reads

We study the election of sequences of committees, where in each of τ\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 respect to the satisfaction per day and per agent. More precisely, we look for egalitarian or equitable committee sequences. While both guarantee that at least x agents per day are satisfied, egalitarian committee sequences ensure that each agent is satisfied in at least y levels while equitable committee sequences ensure that each agent is satisfied in exactly y levels. We analyze the parameterized complexity of finding such committees for the parameters n,m,k,τ,xn,m,k,\tau,x, and y, as well as combinations thereof.


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

March 2023

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41 Reads

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2 Citations

Discrete Applied Mathematics

Matthias Bentert

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René Van Bevern

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Till Fluschnik

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[...]

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Rolf Niedermeier

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.


Overview of our results. “p-NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {NP}}$$\end{document}-h.”, “W[1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {W[1]}}$$\end{document}-h.”, “FPT”, “PK”, and “noPK” respectively abbreviate para-NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {NP}}$$\end{document}-hard, W[1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {W[1]}}$$\end{document}-hard, fixed-parameter tractable, polynomial kernel, and “no polynomial kernel unless NP⊆coNP/poly\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {NP}}\subseteq {\text {coNP}}{/}{\text {poly}}$$\end{document}”. Note that ℓ≤2k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \le 2k$$\end{document} and k≤2ν↓+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\le 2\nu _{\downarrow }+1$$\end{document}
Illustration of Constructions 1 with a illustrating the first snapshot and b illustrating the second snapshot, exemplified for clause C1=(x1∨xj¯∨xi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_1=(x_1 \vee \overline{x_j} \vee x_i)$$\end{document}. The edge {a11,a21}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a_1^1,a_2^1\}$$\end{document} is highlighted in both (a) and (b)
Illustration of Constructions 4 with a showing an odd snapshot and b showing the even snapshot G2ϕ(i,v,j)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{2\phi (i,v,j)}$$\end{document} with edge {aj,w}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a_j,w\}$$\end{document} being present assuming {v,w}∈E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{v,w\}\in E$$\end{document}, and dotted edges may or may not be present (depending on E)
Illustration of Case 1 in the proof of Lemma 2, where |V(Pi+1)\V(Pi)|>k-ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|V(P_{i+1}) {\setminus } V(P_i)| > k-\ell $$\end{document}
Illustration of Constructions 8 with p input instances. a shows a snapshot (V,Er)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(V,E_r)$$\end{document} with r≤log(p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \le \log (p)$$\end{document}. b shows a snapshot (V,Elog(p)+r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(V,E_{\log (p)+r})$$\end{document} for the r-th clause of each input instance. Observe that the green (bright) vertices (including s, t) form a vertex cover of the underlying graph
Multistage s–t Path: Confronting Similarity with Dissimilarity

January 2023

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21 Reads

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10 Citations

Algorithmica

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, 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 in the found path sequence the consecutive s – t paths are “similar”. We measure similarity by the size of the symmetric difference of either the vertex set (vertex-similarity) or the edge set (edge-similarity) of any two consecutive paths. We prove that these two variants of Multistage s – t Path are already NP{\text {NP}} NP -hard for an input sequence of only two snapshots and maximum vertex degree four. Motivated by this fact and natural applications of this scenario e.g. in traffic route planning, we perform a parameterized complexity analysis. Among other results, for both variants, vertex- and edge-similarity, we prove parameterized hardness ( W[1]{\text {W[1]}} W[1] -hardness) regarding the parameter path length (solution size). As a further conceptual investigation, we then modify the multistage model by asking for dissimilar consecutive paths. As one of the main technical results (employing so-called representative sets known from non-temporal settings), we prove that dissimilarity allows for fixed-parameter tractability for the parameter solution size, contrasting with our W[1]-hardness proof of the corresponding similarity case. We also provide partially positive results concerning efficient and effective data reduction (kernelization).


Placing Green Bridges Optimally, with Habitats Inducing Cycles

July 2022

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8 Reads

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1 Citation

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 locations, as well as the habitable vertex set for each species, find the cheapest set of edges such that each species' habitat is connected. We study this problem from a theoretical (algorithms and complexity) and an experimental perspective, while focusing on the case where habitats induce cycles. We prove that the NP-hardness persists in this case even if the graph structure is restricted. If the habitats additionally induce faces in plane graphs however, the problem becomes efficiently solvable. In our empirical evaluation we compare this algorithm as well as ILP formulations for more general variants and an approximation algorithm with another. Our evaluation underlines that each specialization is beneficial in terms of running time, whereas the approximation provides highly competitive solutions in practice.


When Votes Change and Committees Should (Not)

July 2022

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28 Reads

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6 Citations

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 is to find a small committee for each stage of high score. In the conservative model we additionally require that any two consecutive committees have a small symmetric difference. Analogously, in the revolutionary model we require large symmetric differences. We prove both models to be NP-hard even for a constant number of agents, and, based on this, initiate a parameterized complexity analysis for the most natural parameters and combinations thereof. Among other results, we prove both models to be in XP yet W[1]-hard regarding the number of stages, and that being revolutionary seems to be "easier" than being conservative.


An illustrative example with temporal graph G=(G1,G2,G3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {G}=(G_{1},G_{2},G_{3})$\end{document} over the vertex set V={v1,…,v4}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V=\{v_{1},\dots ,v_{4}\}$\end{document}. A solution S=({v2,v3},{v3},{v1,v3}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {S}=(\{v_{2},v_{3}\},\{v_{3}\},\{v_{1},v_{3}\}$\end{document}) for k = 2 and ℓ = 1 is highlighted
Illustrative example of a configuration graph. (a) Temporal graph instance I=(G,k,ℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I=(\mathcal {G},k,\ell )$\end{document} from Fig. 1 with G=(G1,G2,G3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {G}=(G_{1},G_{2},G_{3})$\end{document}, k = 2, and ℓ = 1. (b) Configuration graph of I from (a); a directed s-t path is highlighted corresponding to the solution depicted in Fig. 1
Illustration of Construction 5.1 on an example graph (left-hand side) and the first seven layers of the obtained graph (right-hand side). Dashed vertical lines separate layers, and for each layer all present edges (but only their incident vertices) are depicted. Star-shapes illustrate star graphs with k′+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k^{\prime }+1$\end{document} leaves. Vertices in a solution (layers’ vertex covers) are highlighted
Illustration of Reduction Rule 6.2, exemplified for two vertices u,v and k = 5. Each ellipse for a graph Gi and Gi′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G_{i}^{\prime }$\end{document}, respectively, represents Gi −{u,v} and Gi′−{u,v,wu,wv}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G_{i}^{\prime }-\{u,v,w_{u},w_{v}\}$\end{document}. The vertices wv,wu (gray squares) are introduced by the application of Reduction Rule 6.8. Note that u (v) has a high degree in G1 (G2) and G4
Multistage Vertex Cover

April 2022

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55 Reads

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16 Citations

Theory of Computing Systems

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 Multistage Vertex Cover. Herein, given a temporal graph, the goal is to find for each layer of the temporal graph a small vertex cover and to guarantee that two vertex cover sets of every two consecutive layers differ not too much (specified by a given parameter). We show that, different from classic Vertex Cover and some other dynamic or temporal variants of it, Multistage Vertex Cover is computationally hard even in fairly restricted settings. On the positive side, however, we also spot several fixed-parameter tractability results based on some of themost natural parameterizations.


Citations (44)


... They consider constraints on the extent to which a committee can change, while ensuring a certain level of support remains for the committee. Deltl et al. [15] looked into a similar model, but with the restriction that agents can only approve at most one project per timestep. ...

Reference:

Temporal Elections: Welfare, Strategyproofness, and Proportionality
Algorithmics of Egalitarian versus Equitable Sequences of Committees
  • Citing Conference Paper
  • August 2023

... 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. ...

Polynomial-Time Data Reduction for Weighted Problems Beyond Additive Goal Functions
  • Citing Article
  • March 2023

Discrete Applied Mathematics

... 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 . ...

Multistage s–t Path: Confronting Similarity with Dissimilarity

Algorithmica

... An adjacent line of work by Bredereck et al. [7,8] and Zech et al. [36] look into sequential committee elections where an entire committee is elected at each timestep. They consider constraints on the extent to which a committee can change, while ensuring a certain level of support remains for the committee. ...

When Votes Change and Committees Should (Not)

... Such data can be modeled by a temporal graph, that is, a multi-layer graph in which the layers are ordered linearly [16,17,[30][31][32][33]. The goal in clustering a temporal graph is to find a clustering that slowly evolves over time consistently with the graph [16][17][18][34][35][36]. ...

Multistage Vertex Cover

Theory of Computing Systems

... 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. ...

Feedback Vertex Set on Hamiltonian Graphs
  • Citing Chapter
  • September 2021

Lecture Notes in Computer Science

... 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. ...

Placing Green Bridges Optimally, with a Multivariate Analysis
  • Citing Chapter
  • July 2021

Lecture Notes in Computer Science

... Typically, the solutions are sets of vertices or edges, and the amount of change is measured with the symmetric difference of the sets. Many combinatorial problems have been studied in the multistage setting including matching problems [35,60,61], vertex cover [62], finding paths [63], 2-coloring [64], and others [65][66][67][68][69]. ...

A Multistage View on 2-Satisfiability
  • Citing Chapter
  • May 2021

Lecture Notes in Computer Science

... 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. ...

On approximate data reduction for the Rural Postman Problem: Theory and experiments
  • Citing Article
  • October 2020

Networks