14 Search Results for "Carmi, Paz"

Document
DOI: 10.4230/LIPIcs.WADS.2025.10
On Geodesic Disks Enclosing Many Points

Authors: Prosenjit Bose, Guillermo Esteban, David Orden, Rodrigo I. Silveira, and Tyler Tuttle

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
Let Π(n) be the largest number such that for every set S of n points in a polygon P, there always exist two points x, y ∈ S, where every geodesic disk containing x and y contains Π(n) points of S. We establish upper and lower bounds for Π(n), and show that ⌈n/5⌉ +1 ≤ Π(n) ≤ ⌈n/4⌉ +1. We also show that there always exist two points x, y ∈ S such that every geodesic disk with x and y on its boundary contains at least 16/665(n-2) ≈ ⌈(n-2)/41.6⌉ points both inside and outside the disk. For the special case where the points of S are restricted to be the vertices of a geodesically convex polygon we give a tight bound of ⌈n/3⌉ + 1. We provide the same tight bound when we only consider geodesic disks having x and y as diametral endpoints. Finally, we give a lower bound of ⌈(n-2)/36⌉+2 for the two-colored version of the problem.
Cite as

Prosenjit Bose, Guillermo Esteban, David Orden, Rodrigo I. Silveira, and Tyler Tuttle. On Geodesic Disks Enclosing Many Points. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bose_et_al:LIPIcs.WADS.2025.10,
  author =	{Bose, Prosenjit and Esteban, Guillermo and Orden, David and Silveira, Rodrigo I. and Tuttle, Tyler},
  title =	{{On Geodesic Disks Enclosing Many Points}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.10},
  URN =		{urn:nbn:de:0030-drops-242414},
  doi =		{10.4230/LIPIcs.WADS.2025.10},
  annote =	{Keywords: Enclosing disks, Geodesic disks, Bichromatic}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.26
Geometric Spanners of Bounded Tree-Width

Authors: Kevin Buchin, Carolin Rehs, and Torben Scheele

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Given a point set P in the Euclidean space, a geometric t-spanner G is a graph on P such that for every pair of points, the shortest path in G between those points is at most a factor t longer than the Euclidean distance between those points. The value t ≥ 1 is called the dilation of G. Commonly, the aim is to construct a t-spanner with additional desirable properties. In graph theory, a powerful tool to admit efficient algorithms is bounded tree-width. We therefore investigate the problem of computing geometric spanners with bounded tree-width and small dilation t. Let d be a fixed integer and P ⊂ ℝ^d be a point set with n points. We give a first algorithm to compute an 𝒪(n/k^{d/(d-1)})-spanner on P with tree-width at most k. The dilation obtained by the algorithm is asymptotically worst-case optimal for graphs with tree-width k: We show that there is a set of n points such that every spanner of tree-width k has dilation 𝒪(n/k^{d/(d-1)}). We further prove a tight dependency between tree-width and the number of edges in sparse connected planar graphs, which admits, for point sets in ℝ², a plane spanner with tree-width at most k and small maximum vertex degree. Finally, we show an almost tight bound on the minimum dilation of a spanning tree of n equally spaced points on a circle.
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Kevin Buchin, Carolin Rehs, and Torben Scheele. Geometric Spanners of Bounded Tree-Width. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2025.26,
  author =	{Buchin, Kevin and Rehs, Carolin and Scheele, Torben},
  title =	{{Geometric Spanners of Bounded Tree-Width}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.26},
  URN =		{urn:nbn:de:0030-drops-231786},
  doi =		{10.4230/LIPIcs.SoCG.2025.26},
  annote =	{Keywords: Computational Geometry, Geometric Spanner, Tree-width}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.63
The Maximum Clique Problem in a Disk Graph Made Easy

Authors: J. Mark Keil and Debajyoti Mondal

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
A disk graph is an intersection graph of disks in ℝ². Determining the computational complexity of finding a maximum clique in a disk graph is a long-standing open problem. In 1990, Clark, Colbourn, and Johnson gave a polynomial-time algorithm for computing a maximum clique in a unit disk graph. However, finding a maximum clique when disks are of arbitrary size is widely believed to be a challenging open problem. In this paper, we provide a new perspective to examine adjacencies in a disk graph that helps obtain the following results. - We design an 𝒪^*(n^{2k})-time algorithm, where 𝒪^* hides a polynomial factor, to find a maximum clique in a n-vertex disk graph with k different sizes of radii. This is polynomial for every fixed k, and thus settles the open question for the case when k = 2. - Given a set of n unit disks, we show how to compute a maximum clique inside each possible axis-aligned rectangle determined by the disk centers in O(n⁵log n)-time. This is at least a factor of n^{4/3} faster than applying the fastest known algorithm for finding a maximum clique in a unit disk graph for each rectangle independently. - We give an 𝒪^*(n^{2rk})-time algorithm to find a maximum clique in a n-vertex ball graph with k different sizes of radii where the ball centers lie on r parallel planes. This is polynomial for every fixed k and r, and thus contrasts the previously known NP-hardness result for finding a maximum clique in an arbitrary ball graph. - We design an 𝒪^*(n^{2k})-time algorithm to find a maximum clique in the intersection graph of a set S of n L-visible convex polygons, where k is the number of distinct shapes in S. This contrasts the known hardness result on finding a maximum clique in the intersection graph of unit disks and axis-aligned rectangles.
Cite as

J. Mark Keil and Debajyoti Mondal. The Maximum Clique Problem in a Disk Graph Made Easy. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 63:1-63:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{keil_et_al:LIPIcs.SoCG.2025.63,
  author =	{Keil, J. Mark and Mondal, Debajyoti},
  title =	{{The Maximum Clique Problem in a Disk Graph Made Easy}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{63:1--63:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.63},
  URN =		{urn:nbn:de:0030-drops-232155},
  doi =		{10.4230/LIPIcs.SoCG.2025.63},
  annote =	{Keywords: Geometric Intersection Graphs, Disk Graphs, Ball Graphs, Maximum Clique}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.77
Dynamic Maximum Depth of Geometric Objects

Authors: Subhash Suri, Jie Xue, Xiongxin Yang, and Jiumu Zhu

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Given a set of geometric objects in the plane (rectangles, squares, disks etc.), its maximum depth (or geometric clique) is the largest number of objects with a common intersection. In this paper, we present data structures for dynamically maintaining the maximum depth under insertions and deletions of geometric objects, with sublinear update time. We achieve the following results: - a 1/k-approximate dynamic maximum-depth data structure for (axis-parallel) rectangles with O(n^{1/(k+1)} log n) amortized update time, for any fixed k ∈ ℤ^+. In particular, when k = 1, this gives an exact data structure for rectangles with O(√n log n) amortized update time, almost matching the best known bound for the offline version of the problem. - a (1/2-ε)-approximate dynamic maximum-depth data structure for disks with n^{2/3} log^{O(1)}n amortized update time, for any constant ε > 0. Having exact data structures for disks with sublinear update time is unlikely, since the static maximum-depth problem for disks is 3SUM-hard and thus does not admit subquadratic-time algorithms.
Cite as

Subhash Suri, Jie Xue, Xiongxin Yang, and Jiumu Zhu. Dynamic Maximum Depth of Geometric Objects. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 77:1-77:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{suri_et_al:LIPIcs.SoCG.2025.77,
  author =	{Suri, Subhash and Xue, Jie and Yang, Xiongxin and Zhu, Jiumu},
  title =	{{Dynamic Maximum Depth of Geometric Objects}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{77:1--77:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.77},
  URN =		{urn:nbn:de:0030-drops-232295},
  doi =		{10.4230/LIPIcs.SoCG.2025.77},
  annote =	{Keywords: dynamic algorithms, maximum depth}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.51
Parameterized Geometric Graph Modification with Disk Scaling

Authors: Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
The parameterized analysis of graph modification problems represents the most extensively studied area within Parameterized Complexity. Given a graph G and an integer k ∈ ℕ as input, the goal is to determine whether we can perform at most k operations on G to transform it into a graph belonging to a specified graph class ℱ. Typical operations are combinatorial and include vertex deletions and edge deletions, insertions, and contractions. However, in many real-world scenarios, when the input graph is constrained to be a geometric intersection graph, the modification of the graph is influenced by changes in the geometric properties of the underlying objects themselves, rather than by combinatorial modifications. It raises the question of whether vertex deletions or adjacency modifications are necessarily the most appropriate modification operations for studying modifications of geometric graphs. We propose the study of the disk intersection graph modification through the scaling of disks. This operation is typical in the realm of topology control but has not yet been explored in the context of Parameterized Complexity. We design parameterized algorithms and kernels for modifying to the most basic graph classes: edgeless, connected, and acyclic. Our technical contributions encompass a novel combination of linear programming, branching, and kernelization techniques, along with a fresh application of bidimensionality theory to analyze the area covered by disks, which may have broader applicability.
Cite as

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi. Parameterized Geometric Graph Modification with Disk Scaling. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fomin_et_al:LIPIcs.ITCS.2025.51,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Saurabh, Saket and Zehavi, Meirav},
  title =	{{Parameterized Geometric Graph Modification with Disk Scaling}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{51:1--51:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.51},
  URN =		{urn:nbn:de:0030-drops-226795},
  doi =		{10.4230/LIPIcs.ITCS.2025.51},
  annote =	{Keywords: parameterized algorithms, kernelization, spreading points, distant representatives, unit disk packing}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.1
Planar Bichromatic Bottleneck Spanning Trees

Authors: A. Karim Abu-Affash, Sujoy Bhore, Paz Carmi, and Joseph S. B. Mitchell

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2020.13
Parameterized Study of Steiner Tree on Unit Disk Graphs

Authors: Sujoy Bhore, Paz Carmi, Sudeshna Kolay, and Meirav Zehavi

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.23
Geodesic Obstacle Representation of Graphs

Authors: Prosenjit Bose, Paz Carmi, Vida Dujmovic, Saeed Mehrabi, Fabrizio Montecchiani, Pat Morin, and Luis Fernando Schultz Xavier da Silveira

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.2
Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs

Authors: A. Karim Abu-Affash, Paz Carmi, Anil Maheshwari, Pat Morin, Michiel Smid, and Shakhar Smorodinsky

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.8
Faster Algorithms for some Optimization Problems on Collinear Points

Authors: Ahmad Biniaz, Prosenjit Bose, Paz Carmi, Anil Maheshwari, Ian Munro, and Michiel Smid

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2018.12
Boundary Labeling for Rectangular Diagrams

Authors: Prosenjit Bose, Paz Carmi, J. Mark Keil, Saeed Mehrabi, and Debajyoti Mondal

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

Abstract
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).
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2017.6
Network Optimization on Partitioned Pairs of Points

Authors: Esther M. Arkin, Aritra Banik, Paz Carmi, Gui Citovsky, Su Jia, Matthew J. Katz, Tyler Mayer, and Joseph S. B. Mitchell

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2014.25
Locating Battery Charging Stations to Facilitate Almost Shortest Paths

Authors: Esther M. Arkin, Paz Carmi, Matthew J. Katz, Joseph S. B. Mitchell, and Michael Segal

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/DagSemProc.06481.3
Power Assignment in Radio Networks with Two Power Levels

Authors: Paz Carmi and Matthew Katz

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

Abstract
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.
Cite as

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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