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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8√2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck λ can be converted to a planar bichromatic spanning tree of bottleneck at most 8√2 λ.

A. Karim Abu-Affash, Sujoy Bhore, Paz Carmi, and Joseph S. B. Mitchell. Planar Bichromatic Bottleneck Spanning Trees. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 1:1-1:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{abuaffash_et_al:LIPIcs.ESA.2020.1, author = {Abu-Affash, A. Karim and Bhore, Sujoy and Carmi, Paz and Mitchell, Joseph S. B.}, title = {{Planar Bichromatic Bottleneck Spanning Trees}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {1:1--1:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.1}, URN = {urn:nbn:de:0030-drops-128670}, doi = {10.4230/LIPIcs.ESA.2020.1}, annote = {Keywords: Approximation Algorithms, Bottleneck Spanning Tree, NP-Hardness} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

We study the Steiner Tree problem on unit disk graphs. Given a n vertex unit disk graph G, a subset R⊆ V(G) of t vertices and a positive integer k, the objective is to decide if there exists a tree T in G that spans over all vertices of R and uses at most k vertices from V⧵ R. The vertices of R are referred to as terminals and the vertices of V(G)⧵ R as Steiner vertices. First, we show that the problem is NP-hard. Next, we prove that the Steiner Tree problem on unit disk graphs can be solved in n^{O(√{t+k})} time. We also show that the Steiner Tree problem on unit disk graphs parameterized by k has an FPT algorithm with running time 2^{O(k)}n^{O(1)}. In fact, the algorithms are designed for a more general class of graphs, called clique-grid graphs [Fomin et al., 2019]. We mention that the algorithmic results can be made to work for Steiner Tree on disk graphs with bounded aspect ratio. Finally, we prove that Steiner Tree on disk graphs parameterized by k is W[1]-hard.

Sujoy Bhore, Paz Carmi, Sudeshna Kolay, and Meirav Zehavi. Parameterized Study of Steiner Tree on Unit Disk Graphs. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 13:1-13:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhore_et_al:LIPIcs.SWAT.2020.13, author = {Bhore, Sujoy and Carmi, Paz and Kolay, Sudeshna and Zehavi, Meirav}, title = {{Parameterized Study of Steiner Tree on Unit Disk Graphs}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {13:1--13:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.13}, URN = {urn:nbn:de:0030-drops-122607}, doi = {10.4230/LIPIcs.SWAT.2020.13}, annote = {Keywords: Unit Disk Graphs, FPT, Subexponential exact algorithms, NP-Hardness, W-Hardness} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. The obstacle representation and its plane variant (in which the resulting representation is a plane straight-line embedding of the graph) have been extensively studied with the main objective of minimizing the number of obstacles. Recently, Biedl and Mehrabi [Therese C. Biedl and Saeed Mehrabi, 2017] studied non-blocking grid obstacle representations of graphs in which the vertices of the graph are mapped onto points in the plane while the straight-line segments representing the adjacency between the vertices is replaced by the L_1 (Manhattan) shortest paths in the plane that avoid obstacles.
In this paper, we introduce the notion of geodesic obstacle representations of graphs with the main goal of providing a generalized model, which comes naturally when viewing line segments as shortest paths in the Euclidean plane. To this end, we extend the definition of obstacle representation by allowing some obstacles-avoiding shortest path between the corresponding points in the underlying metric space whenever the vertices are adjacent in the graph. We consider both general and plane variants of geodesic obstacle representations (in a similar sense to obstacle representations) under any polyhedral distance function in R^d as well as shortest path distances in graphs. Our results generalize and unify the notions of obstacle representations, plane obstacle representations and grid obstacle representations, leading to a number of questions on such representations.

Prosenjit Bose, Paz Carmi, Vida Dujmovic, Saeed Mehrabi, Fabrizio Montecchiani, Pat Morin, and Luis Fernando Schultz Xavier da Silveira. Geodesic Obstacle Representation of Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 23:1-23:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bose_et_al:LIPIcs.ICALP.2018.23, author = {Bose, Prosenjit and Carmi, Paz and Dujmovic, Vida and Mehrabi, Saeed and Montecchiani, Fabrizio and Morin, Pat and Silveira, Luis Fernando Schultz Xavier da}, title = {{Geodesic Obstacle Representation of Graphs}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {23:1--23:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.23}, URN = {urn:nbn:de:0030-drops-90274}, doi = {10.4230/LIPIcs.ICALP.2018.23}, annote = {Keywords: Obstacle representation, Grid obstacle representation, Geodesic obstacle representation} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We consider a well studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and an integer d >= 1, in the maximum diameter-bounded subgraph problem (MaxDBS for short), the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For d=1, this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor n^{1-epsilon}, for any epsilon > 0. Moreover, it is known that, for any d >= 2, it is NP-hard to approximate MaxDBS within a factor n^{1/2 - epsilon}, for any epsilon > 0.
In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems and several geometric properties of unit disk graphs.

A. Karim Abu-Affash, Paz Carmi, Anil Maheshwari, Pat Morin, Michiel Smid, and Shakhar Smorodinsky. Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 2:1-2:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{abuaffash_et_al:LIPIcs.SoCG.2018.2, author = {Abu-Affash, A. Karim and Carmi, Paz and Maheshwari, Anil and Morin, Pat and Smid, Michiel and Smorodinsky, Shakhar}, title = {{Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {2:1--2:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.2}, URN = {urn:nbn:de:0030-drops-87152}, doi = {10.4230/LIPIcs.SoCG.2018.2}, annote = {Keywords: Approximation algorithms, maximum diameter-bounded subgraph, unit disk graphs, fractional Helly theorem, VC-dimension} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We propose faster algorithms for the following three optimization problems on n collinear points, i.e., points in dimension one. The first two problems are known to be NP-hard in higher dimensions.
1) Maximizing total area of disjoint disks: In this problem the goal is to maximize the total area of nonoverlapping disks centered at the points. Acharyya, De, and Nandy (2017) presented an O(n^2)-time algorithm for this problem. We present an optimal Theta(n)-time algorithm.
2) Minimizing sum of the radii of client-server coverage: The n points are partitioned into two sets, namely clients and servers. The goal is to minimize the sum of the radii of disks centered at servers such that every client is in some disk, i.e., in the coverage range of some server. Lev-Tov and Peleg (2005) presented an O(n^3)-time algorithm for this problem. We present an O(n^2)-time algorithm, thereby improving the running time by a factor of Theta(n).
3) Minimizing total area of point-interval coverage: The n input points belong to an interval I. The goal is to find a set of disks of minimum total area, covering I, such that every disk contains at least one input point. We present an algorithm that solves this problem in O(n^2) time.

Ahmad Biniaz, Prosenjit Bose, Paz Carmi, Anil Maheshwari, Ian Munro, and Michiel Smid. Faster Algorithms for some Optimization Problems on Collinear Points. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 8:1-8:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{biniaz_et_al:LIPIcs.SoCG.2018.8, author = {Biniaz, Ahmad and Bose, Prosenjit and Carmi, Paz and Maheshwari, Anil and Munro, Ian and Smid, Michiel}, title = {{Faster Algorithms for some Optimization Problems on Collinear Points}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {8:1--8:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.8}, URN = {urn:nbn:de:0030-drops-87219}, doi = {10.4230/LIPIcs.SoCG.2018.8}, annote = {Keywords: collinear points, range assignment} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

Given a set of n points (sites) inside a rectangle R and n points (label locations or ports) on its boundary, a boundary labeling problem seeks ways of connecting every site to a distinct port while achieving different labeling aesthetics. We examine the scenario when the connecting lines (leaders) are drawn as axis-aligned polylines with few bends, every leader lies strictly inside R, no two leaders cross, and the sum of the lengths of all the leaders is minimized. In a k-sided boundary labeling problem, where 1 <= k <= 4, the label locations are located on the k consecutive sides of R.
In this paper we develop an O(n^3 log n)-time algorithm for 2-sided boundary labeling, where the leaders are restricted to have one bend. This improves the previously best known O(n^8 log n)-time algorithm of Kindermann et al. (Algorithmica, 76(1):225-258, 2016). We show the problem is polynomial-time solvable in more general settings such as when the ports are located on more than two sides of R, in the presence of obstacles, and even when the objective is to minimize the total number of bends. Our results improve the previous algorithms on boundary labeling with obstacles, as well as provide the first polynomial-time algorithms for minimizing the total leader length and number of bends for 3- and 4-sided boundary labeling. These results settle a number of open questions on the boundary labeling problems (Wolff, Handbook of Graph Drawing, Chapter 23, Table 23.1, 2014).

Prosenjit Bose, Paz Carmi, J. Mark Keil, Saeed Mehrabi, and Debajyoti Mondal. Boundary Labeling for Rectangular Diagrams. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 12:1-12:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bose_et_al:LIPIcs.SWAT.2018.12, author = {Bose, Prosenjit and Carmi, Paz and Keil, J. Mark and Mehrabi, Saeed and Mondal, Debajyoti}, title = {{Boundary Labeling for Rectangular Diagrams}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.12}, URN = {urn:nbn:de:0030-drops-88386}, doi = {10.4230/LIPIcs.SWAT.2018.12}, annote = {Keywords: Boundary labeling, Dynamic programming, Outerstring graphs} }

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

Given n pairs of points, S = {{p_1, q_1}, {p_2, q_2}, ..., {p_n, q_n}}, in some metric space, we study the problem of two-coloring the points within each pair, red and blue, to optimize the cost of a pair of node-disjoint networks, one over the red points and one over the blue points. In this paper we consider our network structures to be spanning trees, traveling salesman tours or matchings. We consider several different weight functions computed over the network structures induced, as well as several different objective functions. We show that some of these problems are NP-hard, and provide constant factor approximation algorithms in all cases.

Esther M. Arkin, Aritra Banik, Paz Carmi, Gui Citovsky, Su Jia, Matthew J. Katz, Tyler Mayer, and Joseph S. B. Mitchell. Network Optimization on Partitioned Pairs of Points. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 6:1-6:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{arkin_et_al:LIPIcs.ISAAC.2017.6, author = {Arkin, Esther M. and Banik, Aritra and Carmi, Paz and Citovsky, Gui and Jia, Su and Katz, Matthew J. and Mayer, Tyler and Mitchell, Joseph S. B.}, title = {{Network Optimization on Partitioned Pairs of Points}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {6:1--6:12}, 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.2017.6}, URN = {urn:nbn:de:0030-drops-82700}, doi = {10.4230/LIPIcs.ISAAC.2017.6}, annote = {Keywords: Network Optimization, TSP tour, Matching, Spanning Tree, Pairs, Partition, Algorithms, Complexity} }

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**Published in:** OASIcs, Volume 42, 14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2014)

We study a facility location problem motivated by requirements pertaining to the distribution of charging stations for electric vehicles: Place a minimum number of battery charging stations at a subset of nodes of a network, so that battery-powered electric vehicles will be able to move between destinations using "t-spanning" routes, of lengths within a factor t > 1 of the length of a shortest path, while having sufficient charging stations along the way. We give constant-factor approximation algorithms for minimizing the number of charging stations, subject to the t-spanning constraint. We study two versions of the problem, one in which the stations are required to support a single ride (to a single destination), and one in which the stations are to support multiple rides through a sequence of destinations, where the destinations are revealed one at a time.

Esther M. Arkin, Paz Carmi, Matthew J. Katz, Joseph S. B. Mitchell, and Michael Segal. Locating Battery Charging Stations to Facilitate Almost Shortest Paths. In 14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 42, pp. 25-33, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{arkin_et_al:OASIcs.ATMOS.2014.25, author = {Arkin, Esther M. and Carmi, Paz and Katz, Matthew J. and Mitchell, Joseph S. B. and Segal, Michael}, title = {{Locating Battery Charging Stations to Facilitate Almost Shortest Paths}}, booktitle = {14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems}, pages = {25--33}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-75-0}, ISSN = {2190-6807}, year = {2014}, volume = {42}, editor = {Funke, Stefan and Mihal\'{a}k, Mat\'{u}s}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2014.25}, URN = {urn:nbn:de:0030-drops-47500}, doi = {10.4230/OASIcs.ATMOS.2014.25}, annote = {Keywords: approximation algorithms; geometric spanners; transportation networks} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 6481, Geometric Networks and Metric Space Embeddings (2007)

We study the power assignment problem in radio networks, where each radio station can transmit in one of two possible power levels, corresponding to two ranges - short and long. We show that this problem is NP-hard, and present an O(n^2)-time assignment algorithm, such that the number of transmitters that are assigned long range by the algorithm is at most (11/6) times the number of transmitters that are assigned long range by an optimal algorithm. We also present an (9/5)-approximation algorithm for this problem whose running time is considerably higher.
Next, we formulate and study the Minimum-Area Spanning Tree (MAST)problem: Given a set P of n points in the plane, find a spanning tree of P of minimum "area," where the area of a spanning tree T is the area of the union of the n-1 disks whose diameters are the edges in T. We prove that the Euclidean minimum spanning tree of P is a constant-factor approximation for MAST. We then apply this result to obtain constant-factor approximations for the Minimum-Area Range Assignment (MARA) problem, for the Minimum-Area Connected Disk Graph (MACDG) problem, and for the Minimum-Area Tour (MAT) problem. The first problem is a variant of the power assignment problem in radio networks, the second problem is a related natural problem, and the third problem is a variant of the traveling salesman problem.

Paz Carmi and Matthew Katz. Power Assignment in Radio Networks with Two Power Levels. In Geometric Networks and Metric Space Embeddings. Dagstuhl Seminar Proceedings, Volume 6481, pp. 1-18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2007)

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@InProceedings{carmi_et_al:DagSemProc.06481.3, author = {Carmi, Paz and Katz, Matthew}, title = {{Power Assignment in Radio Networks with Two Power Levels}}, booktitle = {Geometric Networks and Metric Space Embeddings}, pages = {1--18}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2007}, volume = {6481}, editor = {Joachim Gudmundsson and Rolf Klein and Giri Narasimhan and Michiel Smid and Alexander Wolff}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06481.3}, URN = {urn:nbn:de:0030-drops-10277}, doi = {10.4230/DagSemProc.06481.3}, annote = {Keywords: Radio networks, power assignment, approximation algorithms, minimum spanning tree, disk graph} }

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