37 Search Results for "de Berg, Mark"


Document
Geometric TSP on Sets

Authors: Henk Alkema and Mark de Berg

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)


Abstract
In One-of-a-Set TSP, also known as the Generalised TSP, the input is a collection 𝒫 : = {P_1, ..., P_r} of sets in a metric space and the goal is to compute a minimum-length tour that visits one element from each set. In the Euclidean variant of this problem, each P_i is a set of points in ℝ^d that is contained in a given hypercube H_i. We investigate how the complexity of Euclidean One-of-a-Set TSP depends on λ, the ply of the set ℋ := {H_1, ..., H_r} of hypercubes (The ply is the smallest λ such that every point in ℝ^d is in at most λ of the hypercubes). Furthermore, we show that the problem can be solved in 2^O(λ^{1/d} n^{1-1/d}) time, where n : = ∑_{i=1}^r |P_i| is the total number of points. Finally, we show that the problem cannot be solved in 2^o(n) time when λ = Θ(n), unless the Exponential Time Hypothesis (ETH) fails. In Rectilinear One-of-a-Cube TSP, the input is a set ℋ of hypercubes in ℝ^d and the goal is to compute a minimum-length rectilinear tour that visits every hypercube. We show that the problem can be solved in 2^O(λ^{1/d} n^{1-1/d} log n) time, where n is the number of hypercubes.

Cite as

Henk Alkema and Mark de Berg. Geometric TSP on Sets. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{alkema_et_al:LIPIcs.ISAAC.2023.6,
  author =	{Alkema, Henk and de Berg, Mark},
  title =	{{Geometric TSP on Sets}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{6:1--6:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.6},
  URN =		{urn:nbn:de:0030-drops-193083},
  doi =		{10.4230/LIPIcs.ISAAC.2023.6},
  annote =	{Keywords: Euclidean TSP, TSP on Sets, Rectilinear TSP, TSP on Neighbourhoods}
}
Document
Clustering in Polygonal Domains

Authors: Mark de Berg, Leyla Biabani, Morteza Monemizadeh, and Leonidas Theocharous

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)


Abstract
We study various clustering problems for a set D of n points in a polygonal domain P under the geodesic distance. We start by studying the discrete k-median problem for D in P. We develop an exact algorithm which runs in time poly(n,m) + n^O(√k), where m is the complexity of the domain. Subsequently, we show that our approach can also be applied to solve the k-center problem with z outliers in the same running time. Next, we turn our attention to approximation algorithms. In particular, we study the k-center problem in a simple polygon and show how to obtain a (1+ε)-approximation algorithm which runs in time 2^{O((k log(k))/ε)} (n log(m) + m). To obtain this, we demonstrate that a previous approach by Bădoiu et al. [Bâdoiu et al., 2002; Bâdoiu and Clarkson, 2003] that works in ℝ^d, carries over to the setting of simple polygons. Finally, we study the 1-center problem in a simple polygon in the presence of z outliers. We show that a coreset C of size O(z) exists, such that the 1-center of C is a 3-approximation of the 1-center of D, when z outliers are allowed. This result is actually more general and carries over to any metric space, which to the best of our knowledge was not known so far. By extending this approach, we show that for the 1-center problem under the Euclidean metric in ℝ², there exists an ε-coreset of size O(z/ε).

Cite as

Mark de Berg, Leyla Biabani, Morteza Monemizadeh, and Leonidas Theocharous. Clustering in Polygonal Domains. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{deberg_et_al:LIPIcs.ISAAC.2023.23,
  author =	{de Berg, Mark and Biabani, Leyla and Monemizadeh, Morteza and Theocharous, Leonidas},
  title =	{{Clustering in Polygonal Domains}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.23},
  URN =		{urn:nbn:de:0030-drops-193252},
  doi =		{10.4230/LIPIcs.ISAAC.2023.23},
  annote =	{Keywords: clustering, geodesic distance, coreset, outliers}
}
Document
Finding Diverse Minimum s-t Cuts

Authors: Mark de Berg, Andrés López Martínez, and Frits Spieksma

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)


Abstract
Recently, many studies have been devoted to finding diverse solutions in classical combinatorial problems, such as Vertex Cover (Baste et al., IJCAI'20), Matching (Fomin et al., ISAAC'20) and Spanning Tree (Hanaka et al., AAAI'21). Finding diverse solutions is important in settings where the user is not able to specify all criteria of the desired solution. Motivated by an application in the field of system identification, we initiate the algorithmic study of k-Diverse Minimum s-t Cuts which, given a directed graph G = (V, E), two specified vertices s,t ∈ V, and an integer k > 0, asks for a collection of k minimum s-t cuts in G that has maximum diversity. We investigate the complexity of the problem for two diversity measures for a collection of cuts: (i) the sum of all pairwise Hamming distances, and (ii) the cardinality of the union of cuts in the collection. We prove that k-Diverse Minimum s-t Cuts can be solved in strongly polynomial time for both diversity measures via submodular function minimization. We obtain this result by establishing a connection between ordered collections of minimum s-t cuts and the theory of distributive lattices. When restricted to finding only collections of mutually disjoint solutions, we provide a more practical algorithm that finds a maximum set of pairwise disjoint minimum s-t cuts. For graphs with small minimum s-t cut, it runs in the time of a single max-flow computation. These results stand in contrast to the problem of finding k diverse global minimum cuts - which is known to be NP-hard even for the disjoint case (Hanaka et al., AAAI'23) - and partially answer a long-standing open question of Wagner (Networks 1990) about improving the complexity of finding disjoint collections of minimum s-t cuts.

Cite as

Mark de Berg, Andrés López Martínez, and Frits Spieksma. Finding Diverse Minimum s-t Cuts. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{deberg_et_al:LIPIcs.ISAAC.2023.24,
  author =	{de Berg, Mark and L\'{o}pez Mart{\'\i}nez, Andr\'{e}s and Spieksma, Frits},
  title =	{{Finding Diverse Minimum s-t Cuts}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{24:1--24:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.24},
  URN =		{urn:nbn:de:0030-drops-193267},
  doi =		{10.4230/LIPIcs.ISAAC.2023.24},
  annote =	{Keywords: S-T MinCut, Diversity, Lattice Theory, Submodular Function Minimization}
}
Document
APPROX
Facility Location in the Sublinear Geometric Model

Authors: Morteza Monemizadeh

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)


Abstract
In the sublinear geometric model, we are provided with an oracle access to a point set P of n points in a bounded discrete space [Δ]², where Δ = n^O(1) is a polynomially bounded number in n. That is, we do not have direct access to the points, but we can make certain types of queries and there is an oracle that responds to our queries. The type of queries that we assume we can make in this paper, are range counting queries where ranges are axis-aligned rectangles (that are basic primitives in database [Srikanta Tirthapura and David P. Woodruff, 2012; Bentley, 1975; Mark de Berg et al., 2008], computational geometry [Pankaj K. Agarwal, 2004; Pankaj K. Agarwal et al., 1996; Boris Aronov et al., 2010; Boris Aronov et al., 2009], and machine learning [Menachem Sadigurschi and Uri Stemmer, 2021; Long and Tan, 1998; Michael J. Kearns and Umesh V. Vazirani, 1995; Michael J. Kearns and Umesh V. Vazirani, 1994]). The oracle then answers these queries by returning the number of points that are in queried ranges. Let {Alg} be an algorithm that (exactly or approximately) solves a problem 𝒫 in the sublinear geometric model. The query complexity of Alg is measured in terms of the number of queries that Alg makes to solve 𝒫. In this paper, we study the complexity of the (uniform) Euclidean facility location problem in the sublinear geometric model. We develop a randomized sublinear algorithm that with high probability, (1+ε)-approximates the cost of the Euclidean facility location problem of the point set P in the sublinear geometric model using Õ(√n) range counting queries. We complement this result by showing that approximating the cost of the Euclidean facility location problem within o(log(n))-factor in the sublinear geometric model using the sampling strategy that we propose for our sublinear algorithm needs Ω̃(n^{1/4}) RangeCount queries. We leave it as an open problem whether such a polynomial lower bound on the number of RangeCount queries exists for any randomized sublinear algorithm that approximates the cost of the facility location problem within a constant factor.

Cite as

Morteza Monemizadeh. Facility Location in the Sublinear Geometric Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{monemizadeh:LIPIcs.APPROX/RANDOM.2023.6,
  author =	{Monemizadeh, Morteza},
  title =	{{Facility Location in the Sublinear Geometric Model}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{6:1--6:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.6},
  URN =		{urn:nbn:de:0030-drops-188315},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.6},
  annote =	{Keywords: Facility Location, Sublinear Geometric Model, Range Counting Queries}
}
Document
APPROX
Stable Approximation Algorithms for Dominating Set and Independent Set

Authors: Mark de Berg, Arpan Sadhukhan, and Frits Spieksma

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)


Abstract
We study Dominating Set and Independent Set for dynamic graphs in the vertex-arrival model. We say that a dynamic algorithm for one of these problems is k-stable when it makes at most k changes to its output independent set or dominating set upon the arrival of each vertex. We study trade-offs between the stability parameter k of the algorithm and the approximation ratio it achieves. We obtain the following results. - We show that there is a constant ε^* > 0 such that any dynamic (1+ε^*)-approximation algorithm for Dominating Set has stability parameter Ω(n), even for bipartite graphs of maximum degree 4. - We present algorithms with very small stability parameters for Dominating Set in the setting where the arrival degree of each vertex is upper bounded by d. In particular, we give a 1-stable (d+1)²-approximation, and a 3-stable (9d/2)-approximation algorithm. - We show that there is a constant ε^* > 0 such that any dynamic (1+ε^*)-approximation algorithm for Independent Set has stability parameter Ω(n), even for bipartite graphs of maximum degree 3. - Finally, we present a 2-stable O(d)-approximation algorithm for Independent Set, in the setting where the average degree of the graph is upper bounded by some constant d at all times.

Cite as

Mark de Berg, Arpan Sadhukhan, and Frits Spieksma. Stable Approximation Algorithms for Dominating Set and Independent Set. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{deberg_et_al:LIPIcs.APPROX/RANDOM.2023.27,
  author =	{de Berg, Mark and Sadhukhan, Arpan and Spieksma, Frits},
  title =	{{Stable Approximation Algorithms for Dominating Set and Independent Set}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{27:1--27:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.27},
  URN =		{urn:nbn:de:0030-drops-188527},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.27},
  annote =	{Keywords: Dynamic algorithms, approximation algorithms, stability, dominating set, independent set}
}
Document
TSP in a Simple Polygon

Authors: Henk Alkema, Mark de Berg, Morteza Monemizadeh, and Leonidas Theocharous

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


Abstract
We study the Traveling Salesman Problem inside a simple polygon. In this problem, which we call tsp in a simple polygon, we wish to compute a shortest tour that visits a given set S of n sites inside a simple polygon P with m edges while staying inside the polygon. This natural problem has, to the best of our knowledge, not been studied so far from a theoretical perspective. It can be solved exactly in poly(n,m) + 2^O(√nlog n) time, using an algorithm by Marx, Pilipczuk, and Pilipczuk (FOCS 2018) for subset tsp as a subroutine. We present a much simpler algorithm that solves tsp in a simple polygon directly and that has the same running time.

Cite as

Henk Alkema, Mark de Berg, Morteza Monemizadeh, and Leonidas Theocharous. TSP in a Simple Polygon. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{alkema_et_al:LIPIcs.ESA.2022.5,
  author =	{Alkema, Henk and de Berg, Mark and Monemizadeh, Morteza and Theocharous, Leonidas},
  title =	{{TSP in a Simple Polygon}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{5:1--5:14},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.5},
  URN =		{urn:nbn:de:0030-drops-169434},
  doi =		{10.4230/LIPIcs.ESA.2022.5},
  annote =	{Keywords: Traveling Salesman Problem, Subexponential algorithms, TSP with obstacles}
}
Document
Computing Smallest Convex Intersecting Polygons

Authors: Antonios Antoniadis, Mark de Berg, Sándor Kisfaludi-Bak, and Antonis Skarlatos

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


Abstract
A polygon C is an intersecting polygon for a set O of objects in ℝ² if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set O of objects. We present an FPTAS for both problems for the case where O is a set of possibly intersecting convex polygons in the plane of total complexity n. Furthermore, we present an exact polynomial-time algorithm for the minimum-perimeter intersecting polygon for the case where O is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum perimeter intersecting polygon of lines or of disjoint segments.

Cite as

Antonios Antoniadis, Mark de Berg, Sándor Kisfaludi-Bak, and Antonis Skarlatos. Computing Smallest Convex Intersecting Polygons. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 9:1-9:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{antoniadis_et_al:LIPIcs.ESA.2022.9,
  author =	{Antoniadis, Antonios and de Berg, Mark and Kisfaludi-Bak, S\'{a}ndor and Skarlatos, Antonis},
  title =	{{Computing Smallest Convex Intersecting Polygons}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{9:1--9:13},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.9},
  URN =		{urn:nbn:de:0030-drops-169470},
  doi =		{10.4230/LIPIcs.ESA.2022.9},
  annote =	{Keywords: convex hull, imprecise points, computational geometry}
}
Document
Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem

Authors: Mark de Berg, Arpan Sadhukhan, and Frits Spieksma

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)


Abstract
Let P be a set of points in ℝ^d (or some other metric space), where each point p ∈ P has an associated transmission range, denoted ρ(p). The range assignment ρ induces a directed communication graph G_{ρ}(P) on P, which contains an edge (p,q) iff |pq| ⩽ ρ(p). In the broadcast range-assignment problem, the goal is to assign the ranges such that G_{ρ}(P) contains an arborescence rooted at a designated root node and the cost ∑_{p ∈ P} ρ(p)² of the assignment is minimized. We study the dynamic version of this problem. In particular, we study trade-offs between the stability of the solution - the number of ranges that are modified when a point is inserted into or deleted from P - and its approximation ratio. To this end we introduce the concept of k-stable algorithms, which are algorithms that modify the range of at most k points when they update the solution. We also introduce the concept of a stable approximation scheme, or SAS for short. A SAS is an update algorithm alg that, for any given fixed parameter ε > 0, is k(ε)-stable and that maintains a solution with approximation ratio 1+ε, where the stability parameter k(ε) only depends on ε and not on the size of P. We study such trade-offs in three settings. - For the problem in ℝ¹, we present a SAS with k(ε) = O(1/ε). Furthermore, we prove that this is tight in the worst case: any SAS for the problem must have k(ε) = Ω(1/ε). We also present algorithms with very small stability parameters: a 1-stable (6+2√5)-approximation algorithm - this algorithm can only handle insertions - a (trivial) 2-stable 2-approximation algorithm, and a 3-stable 1.97-approximation algorithm. - For the problem in 𝕊¹ (that is, when the underlying space is a circle) we prove that no SAS exists. This is in spite of the fact that, for the static problem in 𝕊¹, we prove that an optimal solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in ℝ¹. - For the problem in ℝ², we also prove that no SAS exists, and we present a O(1)-stable O(1)-approximation algorithm. Most results generalize to when the range-assignment cost is ∑_{p ∈ P} ρ(p)^{α}, for some constant α > 1. All omitted theorems and proofs are available in the full version of the paper [Mark de Berg et al., 2021].

Cite as

Mark de Berg, Arpan Sadhukhan, and Frits Spieksma. Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 15:1-15:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{deberg_et_al:LIPIcs.SWAT.2022.15,
  author =	{de Berg, Mark and Sadhukhan, Arpan and Spieksma, Frits},
  title =	{{Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{15:1--15:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.15},
  URN =		{urn:nbn:de:0030-drops-161756},
  doi =		{10.4230/LIPIcs.SWAT.2022.15},
  annote =	{Keywords: Computational geometry, online algorithms, broadcast range assignment, stable approximation schemes}
}
Document
On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem

Authors: Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)


Abstract
We consider the following surveillance problem: Given a set P of n sites in a metric space and a set R of k robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency L of a schedule is the maximum latency of any site, where the latency of a site s is the supremum of the lengths of the time intervals between consecutive visits to s. When k = 1 the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. For k ≥ 2 (which is the version we are interested in) the problem becomes even more challenging; for example, it is not even clear if the decision version of the problem is decidable, in particular in the Euclidean case. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into 𝓁 groups, for some 𝓁 ≤ k, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a 1+ε factor loss in the approximation factor and an O((k/ε) ^k) factor loss in the runtime, for any ε > 0. Our second main result shows that an optimal cyclic solution is a 2(1-1/k)-approximation of the overall optimal solution. Note that for k = 2 this implies that an optimal cyclic solution is optimal overall. We conjecture that this is true for k ≥ 3 as well. The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a (2(1-1/k)+ε)-approximation of the optimal unrestricted (not necessarily cyclic) solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting. Similar results can be obtained by combining our results with other known TSP algorithms in non-Euclidean metrics.

Cite as

Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang. On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{afshani_et_al:LIPIcs.SoCG.2022.2,
  author =	{Afshani, Peyman and de Berg, Mark and Buchin, Kevin and Gao, Jie and L\"{o}ffler, Maarten and Nayyeri, Amir and Raichel, Benjamin and Sarkar, Rik and Wang, Haotian and Yang, Hao-Tsung},
  title =	{{On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{2:1--2:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.2},
  URN =		{urn:nbn:de:0030-drops-160109},
  doi =		{10.4230/LIPIcs.SoCG.2022.2},
  annote =	{Keywords: Approximation, Motion Planning, Scheduling}
}
Document
Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds

Authors: Bahareh Banyassady, Mark de Berg, Karl Bringmann, Kevin Buchin, Henning Fernau, Dan Halperin, Irina Kostitsyna, Yoshio Okamoto, and Stijn Slot

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)


Abstract
We consider the unlabeled motion-planning problem of m unit-disc robots moving in a simple polygonal workspace of n edges. The goal is to find a motion plan that moves the robots to a given set of m target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end. If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time O(n log n + mn + m²) if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance 4 apart, any two target positions are at least distance 4 apart, and any pair of a start and a target positions is at least distance 3 apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary.

Cite as

Bahareh Banyassady, Mark de Berg, Karl Bringmann, Kevin Buchin, Henning Fernau, Dan Halperin, Irina Kostitsyna, Yoshio Okamoto, and Stijn Slot. Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{banyassady_et_al:LIPIcs.SoCG.2022.12,
  author =	{Banyassady, Bahareh and de Berg, Mark and Bringmann, Karl and Buchin, Kevin and Fernau, Henning and Halperin, Dan and Kostitsyna, Irina and Okamoto, Yoshio and Slot, Stijn},
  title =	{{Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{12:1--12:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.12},
  URN =		{urn:nbn:de:0030-drops-160203},
  doi =		{10.4230/LIPIcs.SoCG.2022.12},
  annote =	{Keywords: motion planning, computational geometry, simple polygon}
}
Document
Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model

Authors: Boris Aronov, Mark de Berg, Jean Cardinal, Esther Ezra, John Iacono, and Micha Sharir

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)


Abstract
We present subquadratic algorithms in the algebraic decision-tree model for several 3Sum-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle Δ ∈ C, the number of intersection points between the segments of A and those of B that lie in Δ. The problems considered in this paper have been studied by Chan (2020), who gave algorithms that solve them, in the standard real-RAM model, in O((n²/log²n) log^O(1) log n) time. We present solutions in the algebraic decision-tree model whose cost is O(n^{60/31+ε}), for any ε > 0. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl (2020). A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a "handicap" that turns out to be beneficial for speeding up our algorithm.

Cite as

Boris Aronov, Mark de Berg, Jean Cardinal, Esther Ezra, John Iacono, and Micha Sharir. Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{aronov_et_al:LIPIcs.ISAAC.2021.3,
  author =	{Aronov, Boris and de Berg, Mark and Cardinal, Jean and Ezra, Esther and Iacono, John and Sharir, Micha},
  title =	{{Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.3},
  URN =		{urn:nbn:de:0030-drops-154363},
  doi =		{10.4230/LIPIcs.ISAAC.2021.3},
  annote =	{Keywords: Computational geometry, Algebraic decision-tree model, Polynomial partitioning, Primal-dual range searching, Order types, Point location, Hierarchical partitions}
}
Document
Clique-Based Separators for Geometric Intersection Graphs

Authors: Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, and Leonidas Theocharous

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)


Abstract
Let F be a set of n objects in the plane and let 𝒢^{×}(F) be its intersection graph. A balanced clique-based separator of 𝒢^{×}(F) is a set 𝒮 consisting of cliques whose removal partitions 𝒢^{×}(F) into components of size at most δ n, for some fixed constant δ < 1. The weight of a clique-based separator is defined as ∑_{C ∈ 𝒮}log (|C|+1). Recently De Berg et al. (SICOMP 2020) proved that if S consists of convex fat objects, then 𝒢^{×}(F) admits a balanced clique-based separator of weight O(√n). We extend this result in several directions, obtaining the following results. - Map graphs admit a balanced clique-based separator of weight O(√n), which is tight in the worst case. - Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^{2/3} log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(√n log n). - Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2/3} log n). - Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(√n + r log(n/r)), which is tight in the worst case. These results immediately imply sub-exponential algorithms for MAXIMUM INDEPENDENT SET (and, hence, VERTEX COVER), for FEEDBACK VERTEX SET, and for q-Coloring for constant q in these graph classes.

Cite as

Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, and Leonidas Theocharous. Clique-Based Separators for Geometric Intersection Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{deberg_et_al:LIPIcs.ISAAC.2021.22,
  author =	{de Berg, Mark and Kisfaludi-Bak, S\'{a}ndor and Monemizadeh, Morteza and Theocharous, Leonidas},
  title =	{{Clique-Based Separators for Geometric Intersection Graphs}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{22:1--22:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.22},
  URN =		{urn:nbn:de:0030-drops-154556},
  doi =		{10.4230/LIPIcs.ISAAC.2021.22},
  annote =	{Keywords: Computational geometry, intersection graphs, separator theorems}
}
Document
Maximum-Weight Matching in Sliding Windows and Beyond

Authors: Leyla Biabani, Mark de Berg, and Morteza Monemizadeh

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)


Abstract
We study the maximum-weight matching problem in the sliding-window model. In this model, we are given an adversarially ordered stream of edges of an underlying edge-weighted graph G(V,E), and a parameter L specifying the window size, and we want to maintain an approximation of the maximum-weight matching of the current graph G(t); here G(t) is defined as the subgraph of G consisting of the edges that arrived during the time interval [max(t-L,1),t], where t is the current time. The goal is to do this with Õ(n) space, where n is the number of vertices of G. We present a deterministic (3.5+ε)-approximation algorithm for this problem, thus significantly improving the (6+ε)-approximation algorithm due to Crouch and Stubbs [Michael S. Crouch and Daniel M. Stubbs, 2014]. We also present a generic machinery for approximating subadditve functions in the sliding-window model. A function f is called subadditive if for every disjoint substreams A, B of a stream S it holds that f(AB) ⩽ f(A) + f(B), where AB denotes the concatenation of A and B. We show that given an α-approximation algorithm for a subadditive function f in the insertion-only model we can maintain a (2α+ε)-approximation of f in the sliding-window model. This improves upon recent result Krauthgamer and Reitblat [Robert Krauthgamer and David Reitblat, 2019], who obtained a (2α²+ε)-approximation.

Cite as

Leyla Biabani, Mark de Berg, and Morteza Monemizadeh. Maximum-Weight Matching in Sliding Windows and Beyond. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 73:1-73:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{biabani_et_al:LIPIcs.ISAAC.2021.73,
  author =	{Biabani, Leyla and de Berg, Mark and Monemizadeh, Morteza},
  title =	{{Maximum-Weight Matching in Sliding Windows and Beyond}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{73:1--73:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.73},
  URN =		{urn:nbn:de:0030-drops-155061},
  doi =		{10.4230/LIPIcs.ISAAC.2021.73},
  annote =	{Keywords: maximum-weight matching, sliding-window model, approximation algorithm, and subadditve functions}
}
Document
k-Center Clustering with Outliers in the Sliding-Window Model

Authors: Mark de Berg, Morteza Monemizadeh, and Yu Zhong

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)


Abstract
The k-center problem for a point set P asks for a collection of k congruent balls (that is, balls of equal radius) that together cover all the points in P and whose radius is minimized. The k-center problem with outliers is defined similarly, except that z of the points in P do need not to be covered, for a given parameter z. We study the k-center problem with outliers in data streams in the sliding-window model. In this model we are given a possibly infinite stream P = ⟨ p₁,p₂,p₃,…⟩ of points and a time window of length W, and we want to maintain a small sketch of the set P(t) of points currently in the window such that using the sketch we can approximately solve the problem on P(t). We present the first algorithm for the k-center problem with outliers in the sliding-window model. The algorithm works for the case where the points come from a space of bounded doubling dimension and it maintains a set S(t) such that an optimal solution on S(t) gives a (1+ε)-approximate solution on P(t). The algorithm uses O((kz/ε^d)log σ) storage, where d is the doubling dimension of the underlying space and σ is the spread of the points in the stream. Algorithms providing a (1+ε)-approximation were not even known in the setting without outliers or in the insertion-only setting with outliers. We also present a lower bound showing that any algorithm that provides a (1+ε)-approximation must use Ω((kz/ε)log σ) storage.

Cite as

Mark de Berg, Morteza Monemizadeh, and Yu Zhong. k-Center Clustering with Outliers in the Sliding-Window Model. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{deberg_et_al:LIPIcs.ESA.2021.13,
  author =	{de Berg, Mark and Monemizadeh, Morteza and Zhong, Yu},
  title =	{{k-Center Clustering with Outliers in the Sliding-Window Model}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{13:1--13:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.13},
  URN =		{urn:nbn:de:0030-drops-145945},
  doi =		{10.4230/LIPIcs.ESA.2021.13},
  annote =	{Keywords: Streaming algorithms, k-center problem, sliding window, bounded doubling dimension}
}
Document
Rectilinear Steiner Trees in Narrow Strips

Authors: Henk Alkema and Mark de Berg

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)


Abstract
A rectilinear Steiner tree for a set P of points in ℝ² is a tree that connects the points in P using horizontal and vertical line segments. The goal of {Minimum Rectilinear Steiner Tree} is to find a rectilinear Steiner tree with minimal total length. We investigate how the complexity of {Minimum Rectilinear Steiner Tree} for point sets P inside the strip (-∞,+∞)× [0,δ] depends on the strip width δ. We obtain two main results. - We present an algorithm with running time n^O(√δ) for sparse point sets, that is, point sets where each 1×δ rectangle inside the strip contains O(1) points. - For random point sets, where the points are chosen randomly inside a rectangle of height δ and expected width n, we present an algorithm that is fixed-parameter tractable with respect to δ and linear in n. It has an expected running time of 2^{O(δ √{δ})} n.

Cite as

Henk Alkema and Mark de Berg. Rectilinear Steiner Trees in Narrow Strips. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{alkema_et_al:LIPIcs.SoCG.2021.9,
  author =	{Alkema, Henk and de Berg, Mark},
  title =	{{Rectilinear Steiner Trees in Narrow Strips}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.9},
  URN =		{urn:nbn:de:0030-drops-138081},
  doi =		{10.4230/LIPIcs.SoCG.2021.9},
  annote =	{Keywords: Computational geometry, fixed-parameter tractable algorithms}
}
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