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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Given a weighted graph G, a minimum weight α-spanner is a least-weight subgraph H ⊆ G that preserves minimum distances between all node pairs up to a factor of α. There are many results on heuristics and approximation algorithms, including a recent investigation of their practical performance [Markus Chimani and Finn Stutzenstein, 2022]. Exact approaches, in contrast, have long been denounced as impractical: The first exact ILP (integer linear program) method [Sigurd and Zachariasen, 2004] from 2004 is based on a model with exponentially many path variables, solved via column generation. A second approach [Ahmed et al., 2019], modeling via arc-based multicommodity flow, was presented in 2019. In both cases, only graphs with 40-100 nodes were reported to be solvable.
In this paper, we briefly report on a theoretical comparison between these two models from a polyhedral point of view, and then concentrate on improvements and engineering aspects. We evaluate their performance in a large-scale empirical study. We report that our tuned column generation approach, based on multicriteria shortest path computations, is able to solve instances with over 16 000 nodes within 13 min. Furthermore, now knowing optimal solutions for larger graphs, we are able to investigate the quality of the strongest known heuristic on reasonably sized instances for the first time.

Fritz Bökler, Markus Chimani, Henning Jasper, and Mirko H. Wagner. Exact Minimum Weight Spanners via Column Generation. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bokler_et_al:LIPIcs.ESA.2024.30, author = {B\"{o}kler, Fritz and Chimani, Markus and Jasper, Henning and Wagner, Mirko H.}, title = {{Exact Minimum Weight Spanners via Column Generation}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {30:1--30:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.30}, URN = {urn:nbn:de:0030-drops-211012}, doi = {10.4230/LIPIcs.ESA.2024.30}, annote = {Keywords: Graph spanners, ILP, algorithm engineering, experimental study} }

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**Published in:** LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

A multistage graph problem is a generalization of a traditional graph problem where, instead of a single input graph, we consider a sequence of graphs. We ask for a sequence of solutions, one for each input graph, such that consecutive solutions are as similar as possible. There are several theoretical results on different multistage problems and their complexities, as well as FPT and approximation algorithms. However, there is a severe lack of experimental validation and resulting feedback. Not only are there no algorithmic experiments in literature, we do not even know of any strong set of multistage benchmark instances.
In this paper we want to improve on this situation. We consider the natural problem of multistage shortest path (MSP). First, we propose a rich benchmark set, ranging from synthetic to real-world data, and discuss relevant aspects to ensure non-trivial instances, which is a surprisingly delicate task. Secondly, we present an explorative study on heuristic, approximate, and exact algorithms for the MSP problem from a practical point of view. Our practical findings also inform theoretical research in arguing sensible further directions. For example, based on our study we propose to focus on algorithms for multistage instances that do not rely on 2-stage oracles.

Markus Chimani and Niklas Troost. Multistage Shortest Path: Instances and Practical Evaluation. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chimani_et_al:LIPIcs.SAND.2023.14, author = {Chimani, Markus and Troost, Niklas}, title = {{Multistage Shortest Path: Instances and Practical Evaluation}}, booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)}, pages = {14:1--14:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-275-4}, ISSN = {1868-8969}, year = {2023}, volume = {257}, editor = {Doty, David and Spirakis, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.14}, URN = {urn:nbn:de:0030-drops-179507}, doi = {10.4230/LIPIcs.SAND.2023.14}, annote = {Keywords: Multistage Graphs, Shortest Paths, Experiments} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

A multiplicative α-spanner H is a subgraph of G = (V,E) with the same vertices and fewer edges that preserves distances up to the factor α, i.e., d_H(u,v) ≤ α⋅ d_G(u,v) for all vertices u, v. While many algorithms have been developed to find good spanners in terms of approximation guarantees, no experimental studies comparing different approaches exist. We implemented a rich selection of those algorithms and evaluate them on a variety of instances regarding, e.g., their running time, sparseness, lightness, and effective stretch.

Markus Chimani and Finn Stutzenstein. Spanner Approximations in Practice. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chimani_et_al:LIPIcs.ESA.2022.37, author = {Chimani, Markus and Stutzenstein, Finn}, title = {{Spanner Approximations in Practice}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {37:1--37:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.37}, URN = {urn:nbn:de:0030-drops-169750}, doi = {10.4230/LIPIcs.ESA.2022.37}, annote = {Keywords: Graph spanners, experimental study, algorithm engineering} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

The minimum genus of a graph is an important question in graph theory and a key ingredient in several graph algorithms. However, its computation is NP-hard and turns out to be hard even in practice. Only recently, the first non-trivial approach - based on SAT and ILP (integer linear programming) models - has been presented, but it is unable to successfully tackle graphs of genus larger than 1 in practice.
Herein, we show how to improve the ILP formulation. The crucial ingredients are two-fold. First, we show that instead of modeling rotation schemes explicitly, it suffices to optimize over partitions of the (bidirected) arc set A of the graph. Second, we exploit the cycle structure of the graph, explicitly mapping short closed walks on A to faces in the embedding.
Besides the theoretical advantages of our models, we show their practical strength by a thorough experimental evaluation. Contrary to the previous approach, we are able to quickly solve many instances of genus > 1.

Markus Chimani and Tilo Wiedera. Stronger ILPs for the Graph Genus Problem. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chimani_et_al:LIPIcs.ESA.2019.30, author = {Chimani, Markus and Wiedera, Tilo}, title = {{Stronger ILPs for the Graph Genus Problem}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {30:1--30:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.30}, URN = {urn:nbn:de:0030-drops-111511}, doi = {10.4230/LIPIcs.ESA.2019.30}, annote = {Keywords: algorithm engineering, genus, integer linear programming} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

The NP-hard Maximum Planar Subgraph problem asks for a planar subgraph H of a given graph G such that H has maximum edge cardinality. For more than two decades, the only known non-trivial exact algorithm was based on integer linear programming and Kuratowski's famous planarity criterion. We build upon this approach and present new constraint classes - together with a lifting of the polyhedron - to obtain provably stronger LP-relaxations, and in turn faster algorithms in practice. The new constraints take Euler's polyhedron formula as a starting point and combine it with considering cycles in G. This paper discusses both the theoretical as well as the practical sides of this strengthening.

Markus Chimani and Tilo Wiedera. Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chimani_et_al:LIPIcs.ESA.2018.19, author = {Chimani, Markus and Wiedera, Tilo}, title = {{Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.19}, URN = {urn:nbn:de:0030-drops-94829}, doi = {10.4230/LIPIcs.ESA.2018.19}, annote = {Keywords: algorithm engineering, graph algorithms, integer linear programming, maximum planar subgraph} }

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**Published in:** LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

Given a graph G, the NP-hard Maximum Planar Subgraph problem asks for a planar subgraph of G with the maximum number of edges. The only known non-trivial exact algorithm utilizes Kuratowski's famous planarity criterion and can be formulated as an integer linear program (ILP) or a pseudo-boolean satisfiability problem (PBS). We examine three alternative characterizations of planarity regarding their applicability to model maximum planar subgraphs. For each, we consider both ILP and PBS variants, investigate diverse formulation aspects, and evaluate their practical performance.

Markus Chimani, Ivo Hedtke, and Tilo Wiedera. Exact Algorithms for the Maximum Planar Subgraph Problem: New Models and Experiments. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chimani_et_al:LIPIcs.SEA.2018.22, author = {Chimani, Markus and Hedtke, Ivo and Wiedera, Tilo}, title = {{Exact Algorithms for the Maximum Planar Subgraph Problem: New Models and Experiments}}, booktitle = {17th International Symposium on Experimental Algorithms (SEA 2018)}, pages = {22:1--22:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-070-5}, ISSN = {1868-8969}, year = {2018}, volume = {103}, editor = {D'Angelo, Gianlorenzo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.22}, URN = {urn:nbn:de:0030-drops-89572}, doi = {10.4230/LIPIcs.SEA.2018.22}, annote = {Keywords: maximum planar subgraph, integer linear programming, pseudo boolean satisfiability, graph drawing, algorithm engineering} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

The crossing number is the smallest number of pairwise edge crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions, like treewidth or pathwidth.
In this paper, we for the first time show that crossing number is tractable (even in linear time) for maximal graphs of bounded pathwidth 3. The technique also shows that the crossing number and the rectilinear (a.k.a. straight-line) crossing number are identical for this graph class, and that we require only an O(n)xO(n)-grid to achieve such a drawing.
Our techniques can further be extended to devise a 2-approximation for general graphs with pathwidth 3, and a 4w^3-approximation for maximal graphs of pathwidth w. This is a constant approximation for bounded pathwidth graphs.

Therese Biedl, Markus Chimani, Martin Derka, and Petra Mutzel. Crossing Number for Graphs with Bounded~Pathwidth. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{biedl_et_al:LIPIcs.ISAAC.2017.13, author = {Biedl, Therese and Chimani, Markus and Derka, Martin and Mutzel, Petra}, title = {{Crossing Number for Graphs with Bounded\textasciitildePathwidth}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.13}, URN = {urn:nbn:de:0030-drops-82570}, doi = {10.4230/LIPIcs.ISAAC.2017.13}, annote = {Keywords: Crossing Number, Graphs with Bounded Pathwidth} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Formally, approaches based on mathematical programming are able to find provably optimal solutions.
However, the demands on a verifiable formal proof are typically much higher than the guarantees
we can sensibly attribute to implementations of mathematical programs. We consider this in the context of the crossing number problem, one of the most prominent problems in topological graph theory. The problem asks for the minimum number of edge crossings in any drawing of a given graph. Graph-theoretic proofs for this problem are known to be notoriously hard to obtain. At the same time, proofs even for very specific graphs are often of interest in crossing number research, as they can, e.g., form the basis for inductive proofs.
We propose a system to automatically generate a formal proof based on an ILP computation. Such a proof is (relatively) easily verifiable, and does not require the understanding of any complex ILP codes. As such, we hope our proof system may serve as a showcase for the necessary steps and central design goals of how to establish formal proof systems based on mathematical programming formulations.

Markus Chimani and Tilo Wiedera. An ILP-based Proof System for the Crossing Number Problem. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chimani_et_al:LIPIcs.ESA.2016.29, author = {Chimani, Markus and Wiedera, Tilo}, title = {{An ILP-based Proof System for the Crossing Number Problem}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {29:1--29:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.29}, URN = {urn:nbn:de:0030-drops-63803}, doi = {10.4230/LIPIcs.ESA.2016.29}, annote = {Keywords: automatic formal proof, crossing number, integer linear programming} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Let G be a connected planar (but not yet embedded) graph and F a set of additional edges not in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. An optimal solution to this problem is known to approximate the crossing number of the graph G+F.
Finding an exact solution to MEI is NP-hard for general F, but linear time solvable for the special case of |F|=1 [Gutwenger et al, SODA 2001/Algorithmica] and polynomial time solvable when all of F are incident to a new vertex [Chimani et al, SODA 2009]. The complexity for general F but with constant k=|F| was open, but algorithms both with relative and absolute approximation guarantees have been presented [Chuzhoy et al, SODA 2011], [Chimani-Hlineny, ICALP 2011]. We show that the problem is fixed parameter tractable (FPT) in k for biconnected G, or if the cut vertices of G have bounded degrees. We give the first exact algorithm for this problem; it requires only O(|V(G)|) time for any constant k.

Markus Chimani and Petr Hlinený. Inserting Multiple Edges into a Planar Graph. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chimani_et_al:LIPIcs.SoCG.2016.30, author = {Chimani, Markus and Hlinen\'{y}, Petr}, title = {{Inserting Multiple Edges into a Planar Graph}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {30:1--30:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.30}, URN = {urn:nbn:de:0030-drops-59223}, doi = {10.4230/LIPIcs.SoCG.2016.30}, annote = {Keywords: crossing number, edge insertion, parameterized complexity, path homotopy, funnel algorithm} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

In a directed graph G with non-correlated edge lengths and costs, the network design problem with bounded distances asks for a cost-minimal spanning subgraph subject to a length bound for all node pairs. We give a bi-criteria (2+\varepsilon,O(n^{0.5+\varepsilon}))-approximation for this problem. This improves on the currently best known linear approximation bound, at the cost of violating the distance bound by a factor of at most 2+\varepsilon.
In the course of proving this result, the related problem of directed shallow-light Steiner trees arises as a subproblem. In the context of directed graphs, approximations to this problem have been elusive. We present the first non-trivial result by proposing a (1+\varepsilon,O(|R|^{\varepsilon}))-ap\-proximation, where R is the set of terminals.
Finally, we show how to apply our results to obtain an (\alpha+\varepsilon,O(n^{0.5+\varepsilon}))-approximation for light-weight directed \alpha-spanners. For this, no non-trivial approximation algorithm has been known before. All running times depends on n and
\varepsilon and are polynomial in n for any fixed \varepsilon>0.

Markus Chimani and Joachim Spoerhase. Network Design Problems with Bounded Distances via Shallow-Light Steiner Trees. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 238-248, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chimani_et_al:LIPIcs.STACS.2015.238, author = {Chimani, Markus and Spoerhase, Joachim}, title = {{Network Design Problems with Bounded Distances via Shallow-Light Steiner Trees}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {238--248}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.238}, URN = {urn:nbn:de:0030-drops-49170}, doi = {10.4230/LIPIcs.STACS.2015.238}, annote = {Keywords: network design, approximation algorithm, shallow-light spanning trees, spanners} }

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