28 Search Results for "Welzl, Emo"

Document
DOI: 10.4230/LIPIcs.GD.2024.9
Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs

Authors: Adrian Dumitrescu and János Pach

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
A complete geometric graph consists of a set P of n points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a positive constant c < 1, such that every complete geometric graph on n points can be partitioned into at most cn plane graphs (that is, noncrossing subgraphs). We answer this question in the affirmative in the special case where the underlying point set P is dense, which means that the ratio between the maximum and the minimum distances in P is of the order of Θ(√n).
Cite as

Adrian Dumitrescu and János Pach. Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 9:1-9:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dumitrescu_et_al:LIPIcs.GD.2024.9,
  author =	{Dumitrescu, Adrian and Pach, J\'{a}nos},
  title =	{{Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{9:1--9:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.9},
  URN =		{urn:nbn:de:0030-drops-212939},
  doi =		{10.4230/LIPIcs.GD.2024.9},
  annote =	{Keywords: Convexity, complete geometric Graph, crossing Family, plane Subgraph}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.19
Weakly Leveled Planarity with Bounded Span

Authors: Michael A. Bekos, Giordano Da Lozzo, Fabrizio Frati, Siddharth Gupta, Philipp Kindermann, Giuseppe Liotta, Ignaz Rutter, and Ioannis G. Tollis

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
This paper studies planar drawings of graphs in which each vertex is represented as a point along a sequence of horizontal lines, called levels, and each edge is either a horizontal segment or a strictly y-monotone curve. A graph is s-span weakly leveled planar if it admits such a drawing where the edges have span at most s; the span of an edge is the number of levels it touches minus one. We investigate the problem of computing s-span weakly leveled planar drawings from both the computational and the combinatorial perspectives. We prove the problem to be para-NP-hard with respect to its natural parameter s and investigate its complexity with respect to widely used structural parameters. We show the existence of a polynomial-size kernel with respect to vertex cover number and prove that the problem is FPT when parameterized by treedepth. We also present upper and lower bounds on the span for various graph classes. Notably, we show that cycle trees, a family of 2-outerplanar graphs generalizing Halin graphs, are Θ(log n)-span weakly leveled planar and 4-span weakly leveled planar when 3-connected. As a byproduct of these combinatorial results, we obtain improved bounds on the edge-length ratio of the graph families under consideration.
Cite as

Michael A. Bekos, Giordano Da Lozzo, Fabrizio Frati, Siddharth Gupta, Philipp Kindermann, Giuseppe Liotta, Ignaz Rutter, and Ioannis G. Tollis. Weakly Leveled Planarity with Bounded Span. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bekos_et_al:LIPIcs.GD.2024.19,
  author =	{Bekos, Michael A. and Da Lozzo, Giordano and Frati, Fabrizio and Gupta, Siddharth and Kindermann, Philipp and Liotta, Giuseppe and Rutter, Ignaz and Tollis, Ioannis G.},
  title =	{{Weakly Leveled Planarity with Bounded Span}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.19},
  URN =		{urn:nbn:de:0030-drops-213035},
  doi =		{10.4230/LIPIcs.GD.2024.19},
  annote =	{Keywords: Leveled planar graphs, edge span, graph drawing, edge-length ratio}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.24
Upward Pointset Embeddings of Planar st-Graphs

Authors: Carlos Alegría, Susanna Caroppo, Giordano Da Lozzo, Marco D'Elia, Giuseppe Di Battista, Fabrizio Frati, Fabrizio Grosso, and Maurizio Patrignani

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
We study upward pointset embeddings (UPSEs) of planar st-graphs. Let G be a planar st-graph and let S ⊂ ℝ² be a pointset with |S| = |V(G)|. An UPSE of G on S is an upward planar straight-line drawing of G that maps the vertices of G to the points of S. We consider both the problem of testing the existence of an UPSE of G on S (UPSE Testing) and the problem of enumerating all UPSEs of G on S. We prove that UPSE Testing is NP-complete even for st-graphs that consist of a set of directed st-paths sharing only s and t. On the other hand, for n-vertex planar st-graphs whose maximum st-cutset has size k, we prove that it is possible to solve UPSE Testing in 𝒪(n^{4k}) time with 𝒪(n^{3k}) space, and to enumerate all UPSEs of G on S with 𝒪(n) worst-case delay, using 𝒪(k n^{4k} log n) space, after 𝒪(k n^{4k} log n) set-up time. Moreover, for an n-vertex st-graph whose underlying graph is a cycle, we provide a necessary and sufficient condition for the existence of an UPSE on a given poinset, which can be tested in 𝒪(n log n) time. Related to this result, we give an algorithm that, for a set S of n points, enumerates all the non-crossing monotone Hamiltonian cycles on S with 𝒪(n) worst-case delay, using 𝒪(n²) space, after 𝒪(n²) set-up time.
Cite as

Carlos Alegría, Susanna Caroppo, Giordano Da Lozzo, Marco D'Elia, Giuseppe Di Battista, Fabrizio Frati, Fabrizio Grosso, and Maurizio Patrignani. Upward Pointset Embeddings of Planar st-Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alegria_et_al:LIPIcs.GD.2024.24,
  author =	{Alegr{\'\i}a, Carlos and Caroppo, Susanna and Da Lozzo, Giordano and D'Elia, Marco and Di Battista, Giuseppe and Frati, Fabrizio and Grosso, Fabrizio and Patrignani, Maurizio},
  title =	{{Upward Pointset Embeddings of Planar st-Graphs}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{24:1--24:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.24},
  URN =		{urn:nbn:de:0030-drops-213082},
  doi =		{10.4230/LIPIcs.GD.2024.24},
  annote =	{Keywords: Upward pointset embeddings, planar st-graphs, st-cutset, non-crossing monotone Hamiltonian cycles, graph drawing enumeration}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.39
List Homomorphisms by Deleting Edges and Vertices: Tight Complexity Bounds for Bounded-Treewidth Graphs

Authors: Barış Can Esmer, Jacob Focke, Dániel Marx, and Paweł Rzążewski

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

Abstract
The goal of this paper is to investigate a family of optimization problems arising from list homomorphisms, and to understand what the best possible algorithms are if we restrict the problem to bounded-treewidth graphs. Given graphs G, H, and lists L(v) ⊆ V(H) for every v ∈ V(G), a list homomorphism from (G,L) to H is a function f:V(G) → V(H) that preserves the edges (i.e., uv ∈ E(G) implies f(u)f(v) ∈ E(H)) and respects the lists (i.e., f(v) ∈ L(v)). The graph H may have loops. For a fixed H, the input of the optimization problem LHomVD(H) is a graph G with lists L(v), and the task is to find a set X of vertices having minimum size such that (G-X,L) has a list homomorphism to H. We define analogously the edge-deletion variant LHomED(H), where we have to delete as few edges as possible from G to obtain a graph that has a list homomorphism. This expressive family of problems includes members that are essentially equivalent to fundamental problems such as Vertex Cover, Max Cut, Odd Cycle Transversal, and Edge/Vertex Multiway Cut. For both variants, we first characterize those graphs H that make the problem polynomial-time solvable and show that the problem is NP-hard for every other fixed H. Second, as our main result, we determine for every graph H for which the problem is NP-hard, the smallest possible constant c_H such that the problem can be solved in time c^t_H⋅ n^{𝒪(1)} if a tree decomposition of G having width t is given in the input. Let i(H) be the maximum size of a set of vertices in H that have pairwise incomparable neighborhoods. For the vertex-deletion variant LHomVD(H), we show that the smallest possible constant is i(H)+1 for every H: - Given a tree decomposition of width t of G, LHomVD(H) can be solved in time (i(H)+1)^t⋅ n^{𝒪(1)}. - For any ε > 0 and H, an (i(H)+1-ε)^t⋅ n^{𝒪(1)} algorithm would violate the Strong Exponential-Time Hypothesis (SETH). The situation is more complex for the edge-deletion version. For every H, one can solve LHomED(H) in time i(H)^t⋅ n^{𝒪(1)} if a tree decomposition of width t is given. However, the existence of a specific type of decomposition of H shows that there are graphs H where LHomED(H) can be solved significantly more efficiently and the best possible constant can be arbitrarily smaller than i(H). Nevertheless, we determine this best possible constant and (assuming the SETH) prove tight bounds for every fixed H.
Cite as

Barış Can Esmer, Jacob Focke, Dániel Marx, and Paweł Rzążewski. List Homomorphisms by Deleting Edges and Vertices: Tight Complexity Bounds for Bounded-Treewidth Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{canesmer_et_al:LIPIcs.ESA.2024.39,
  author =	{Can Esmer, Bar{\i}\c{s} and Focke, Jacob and Marx, D\'{a}niel and Rz\k{a}\.{z}ewski, Pawe{\l}},
  title =	{{List Homomorphisms by Deleting Edges and Vertices: Tight Complexity Bounds for Bounded-Treewidth Graphs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{39:1--39:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.39},
  URN =		{urn:nbn:de:0030-drops-211103},
  doi =		{10.4230/LIPIcs.ESA.2024.39},
  annote =	{Keywords: Graph Homomorphism, List Homomorphism, Vertex Deletion, Edge Deletion, Multiway Cut, Parameterized Complexity, Tight Bounds, Treewidth, SETH}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.51
Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs

Authors: Lech Duraj, Filip Konieczny, and Krzysztof Potępa

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

Abstract
We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs. We build on the work of Ducoffe et al. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become simpler and faster, working in 𝒪{(k ⋅ n^{1-1/d} ⋅ m ⋅ polylog(n))} time complexity for the graph on n vertices and m edges, where k is the diameter and d is the distance VC-dimension of the graph. Furthermore, it allows us to use the improved technique in more general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by intersection. Applying our approach for these graphs, we partially answer a question posed by Bringmann et al. [SoCG'22], finding an 𝒪{(n^{7/4} ⋅ polylog(n))} parameterized diameter algorithm for unit square intersection graph of size n, as well as a more general algorithm for convex polygon intersection graphs.
Cite as

Lech Duraj, Filip Konieczny, and Krzysztof Potępa. Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{duraj_et_al:LIPIcs.ESA.2024.51,
  author =	{Duraj, Lech and Konieczny, Filip and Pot\k{e}pa, Krzysztof},
  title =	{{Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{51:1--51:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.51},
  URN =		{urn:nbn:de:0030-drops-211229},
  doi =		{10.4230/LIPIcs.ESA.2024.51},
  annote =	{Keywords: Graph Diameter, Geometric Intersection Graphs, Vapnik-Chervonenkis Dimension}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.12
Universal Optimization for Non-Clairvoyant Subadditive Joint Replenishment

Authors: Tomer Ezra, Stefano Leonardi, Michał Pawłowski, Matteo Russo, and Seeun William Umboh

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
The online joint replenishment problem (JRP) is a fundamental problem in the area of online problems with delay. Over the last decade, several works have studied generalizations of JRP with different cost functions for servicing requests. Most prior works on JRP and its generalizations have focused on the clairvoyant setting. Recently, Touitou [Noam Touitou, 2023] developed a non-clairvoyant framework that provided an O(√{n log n}) upper bound for a wide class of generalized JRP, where n is the number of request types. We advance the study of non-clairvoyant algorithms by providing a simpler, modular framework that matches the competitive ratio established by Touitou for the same class of generalized JRP. Our key insight is to leverage universal algorithms for Set Cover to approximate arbitrary monotone subadditive functions using a simple class of functions termed disjoint. This allows us to reduce the problem to several independent instances of the TCP Acknowledgement problem, for which a simple 2-competitive non-clairvoyant algorithm is known. The modularity of our framework is a major advantage as it allows us to tailor the reduction to specific problems and obtain better competitive ratios. In particular, we obtain tight O(√n)-competitive algorithms for two significant problems: Multi-Level Aggregation and Weighted Symmetric Subadditive Joint Replenishment. We also show that, in contrast, Touitou’s algorithm is Ω(√{n log n})-competitive for both of these problems.
Cite as

Tomer Ezra, Stefano Leonardi, Michał Pawłowski, Matteo Russo, and Seeun William Umboh. Universal Optimization for Non-Clairvoyant Subadditive Joint Replenishment. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 12:1-12:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ezra_et_al:LIPIcs.APPROX/RANDOM.2024.12,
  author =	{Ezra, Tomer and Leonardi, Stefano and Paw{\l}owski, Micha{\l} and Russo, Matteo and Umboh, Seeun William},
  title =	{{Universal Optimization for Non-Clairvoyant Subadditive Joint Replenishment}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{12:1--12:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.12},
  URN =		{urn:nbn:de:0030-drops-210050},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.12},
  annote =	{Keywords: Set Cover, Joint Replenishment, TCP-Acknowledgment, Subadditive Function Approximation, Multi-Level Aggregation}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.9
Quantum Algorithms for Hopcroft’s Problem

Authors: Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
In this work we study quantum algorithms for Hopcroft’s problem which is a fundamental problem in computational geometry. Given n points and n lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in O(n^{4/3}) time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity Õ(n^{5/6}). The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.
Cite as

Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs. Quantum Algorithms for Hopcroft’s Problem. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{andrejevs_et_al:LIPIcs.MFCS.2024.9,
  author =	{Andrejevs, Vladimirs and Belovs, Aleksandrs and Vihrovs, Jevg\={e}nijs},
  title =	{{Quantum Algorithms for Hopcroft’s Problem}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.9},
  URN =		{urn:nbn:de:0030-drops-205653},
  doi =		{10.4230/LIPIcs.MFCS.2024.9},
  annote =	{Keywords: Quantum algorithms, Quantum walks, Computational Geometry}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.11
Switching Classes: Characterization and Computation

Authors: Dhanyamol Antony, Yixin Cao, Sagartanu Pal, and R. B. Sandeep

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
In a graph, the switching operation reverses adjacencies between a subset of vertices and the others. For a hereditary graph class 𝒢, we are concerned with the maximum subclass and the minimum superclass of 𝒢 that are closed under switching. We characterize the maximum subclass for many important classes 𝒢, and prove that it is finite when 𝒢 is minor-closed and omits at least one graph. For several graph classes, we develop polynomial-time algorithms to recognize the minimum superclass. We also show that the recognition of the superclass is NP-hard for H-free graphs when H is a sufficiently long path or cycle, and it cannot be solved in subexponential time assuming the Exponential Time Hypothesis.
Cite as

Dhanyamol Antony, Yixin Cao, Sagartanu Pal, and R. B. Sandeep. Switching Classes: Characterization and Computation. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{antony_et_al:LIPIcs.MFCS.2024.11,
  author =	{Antony, Dhanyamol and Cao, Yixin and Pal, Sagartanu and Sandeep, R. B.},
  title =	{{Switching Classes: Characterization and Computation}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.11},
  URN =		{urn:nbn:de:0030-drops-205678},
  doi =		{10.4230/LIPIcs.MFCS.2024.11},
  annote =	{Keywords: Switching, Graph modification, Minor-closed graph class, Hereditary graph class}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.26
Capturing the Shape of a Point Set with a Line Segment

Authors: Nathan van Beusekom, Marc van Kreveld, Max van Mulken, Marcel Roeloffzen, Bettina Speckmann, and Jules Wulms

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Detecting location-correlated groups in point sets is an important task in a wide variety of applications areas. In addition to merely detecting such groups, the group’s shape carries meaning as well. In this paper, we represent a group’s shape using a simple geometric object, a line segment. Specifically, given a radius r, we say a line segment is representative of a point set P of n points if it is within distance r of each point p ∈ P. We aim to find the shortest such line segment. This problem is equivalent to stabbing a set of circles of radius r using the shortest line segment. We describe an algorithm to find the shortest representative segment in O(n log h + h log³h) time, where h is the size of the convex hull of P. Additionally, we show how to maintain a stable approximation of the shortest representative segment when the points in P move.
Cite as

Nathan van Beusekom, Marc van Kreveld, Max van Mulken, Marcel Roeloffzen, Bettina Speckmann, and Jules Wulms. Capturing the Shape of a Point Set with a Line Segment. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vanbeusekom_et_al:LIPIcs.MFCS.2024.26,
  author =	{van Beusekom, Nathan and van Kreveld, Marc and van Mulken, Max and Roeloffzen, Marcel and Speckmann, Bettina and Wulms, Jules},
  title =	{{Capturing the Shape of a Point Set with a Line Segment}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{26:1--26:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.26},
  URN =		{urn:nbn:de:0030-drops-205820},
  doi =		{10.4230/LIPIcs.MFCS.2024.26},
  annote =	{Keywords: Shape descriptor, Stabbing, Rotating calipers}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.32
Symmetric-Difference (Degeneracy) and Signed Tree Models

Authors: Édouard Bonnet, Julien Duron, John Sylvester, and Viktor Zamaraev

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
We introduce a dense counterpart of graph degeneracy, which extends the recently-proposed invariant symmetric difference. We say that a graph has sd-degeneracy (for symmetric-difference degeneracy) at most d if it admits an elimination order of its vertices where a vertex u can be removed whenever it has a d-twin, i.e., another vertex v such that at most d vertices outside {u,v} are neighbors of exactly one of u, v. The family of graph classes of bounded sd-degeneracy is a superset of that of graph classes of bounded degeneracy or of bounded flip-width, and more generally, of bounded symmetric difference. Unlike most graph parameters, sd-degeneracy is not hereditary: it may be strictly smaller on a graph than on some of its induced subgraphs. In particular, every n-vertex graph is an induced subgraph of some O(n²)-vertex graph of sd-degeneracy 1. In spite of this and the breadth of classes of bounded sd-degeneracy, we devise Õ(√n)-bit adjacency labeling schemes for them, which are optimal up to the hidden polylogarithmic factor. This is attained on some even more general classes, consisting of graphs G whose vertices bijectively map to the leaves of a tree T, where transversal edges and anti-edges added to T define the edge set of G. We call such graph representations signed tree models as they extend the so-called tree models (or twin-decompositions) developed in the context of twin-width, by adding transversal anti-edges. While computing the degeneracy of a graph takes linear time, we show that determining its symmetric difference is para-co-NP-complete. This may seem surprising as symmetric difference can serve as a short-sighted first approximation of twin-width, whose computation is para-NP-complete. Indeed, we show that deciding if the symmetric difference of an input graph is at most 8 is co-NP-complete. We also show that deciding if the sd-degeneracy is at most 6 is NP-complete, contrasting with the symmetric difference.
Cite as

Édouard Bonnet, Julien Duron, John Sylvester, and Viktor Zamaraev. Symmetric-Difference (Degeneracy) and Signed Tree Models. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bonnet_et_al:LIPIcs.MFCS.2024.32,
  author =	{Bonnet, \'{E}douard and Duron, Julien and Sylvester, John and Zamaraev, Viktor},
  title =	{{Symmetric-Difference (Degeneracy) and Signed Tree Models}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{32:1--32:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.32},
  URN =		{urn:nbn:de:0030-drops-205886},
  doi =		{10.4230/LIPIcs.MFCS.2024.32},
  annote =	{Keywords: symmetric difference, degeneracy, adjacency labeling schemes, NP-hardness}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.8
Practical Computation of Graph VC-Dimension

Authors: David Coudert, Mónika Csikós, Guillaume Ducoffe, and Laurent Viennot

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
For any set system ℋ = (V,ℛ), ℛ ⊆ 2^V, a subset S ⊆ V is called shattered if every S' ⊆ S results from the intersection of S with some set in ℛ. The VC-dimension of ℋ is the size of a largest shattered set in V. In this paper, we focus on the problem of computing the VC-dimension of graphs. In particular, given a graph G = (V,E), the VC-dimension of G is defined as the VC-dimension of (V, N), where N contains each subset of V that can be obtained as the closed neighborhood of some vertex v ∈ V in G. Our main contribution is an algorithm for computing the VC-dimension of any graph, whose effectiveness is shown through experiments on various types of practical graphs, including graphs with millions of vertices. A key aspect of its efficiency resides in the fact that practical graphs have small VC-dimension, up to 8 in our experiments. As a side-product, we present several new bounds relating the graph VC-dimension to other classical graph theoretical notions. We also establish the W[1]-hardness of the graph VC-dimension problem by extending a previous result for arbitrary set systems.
Cite as

David Coudert, Mónika Csikós, Guillaume Ducoffe, and Laurent Viennot. Practical Computation of Graph VC-Dimension. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{coudert_et_al:LIPIcs.SEA.2024.8,
  author =	{Coudert, David and Csik\'{o}s, M\'{o}nika and Ducoffe, Guillaume and Viennot, Laurent},
  title =	{{Practical Computation of Graph VC-Dimension}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.8},
  URN =		{urn:nbn:de:0030-drops-203731},
  doi =		{10.4230/LIPIcs.SEA.2024.8},
  annote =	{Keywords: VC-dimension, graph, algorithm}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.32
Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents

Authors: Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We provide polynomial-time reductions between three search problems from three distinct areas: the P-matrix linear complementarity problem (P-LCP), finding the sink of a unique sink orientation (USO), and a variant of the α-Ham Sandwich problem. For all three settings, we show that "two choices are enough", meaning that the general non-binary version of the problem can be reduced in polynomial time to the binary version. This specifically means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are equivalent to cube USOs. These results are obtained by showing that both the P-LCP and our α-Ham Sandwich variant are equivalent to a new problem we introduce, P-Lin-Bellman. This problem can be seen as a new tool for formulating problems as P-LCPs.
Cite as

Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber. Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{borzechowski_et_al:LIPIcs.ICALP.2024.32,
  author =	{Borzechowski, Michaela and Fearnley, John and Gordon, Spencer and Savani, Rahul and Schnider, Patrick and Weber, Simon},
  title =	{{Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.32},
  URN =		{urn:nbn:de:0030-drops-201751},
  doi =		{10.4230/LIPIcs.ICALP.2024.32},
  annote =	{Keywords: P-LCP, Unique Sink Orientation, \alpha-Ham Sandwich, search complexity, TFNP, UEOPL}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.23
A Canonical Tree Decomposition for Chirotopes

Authors: Mathilde Bouvel, Valentin Feray, Xavier Goaoc, and Florent Koechlin

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets, and adapts in some sense the modular decomposition of graphs in the world of chirotopes. The associated tree always exists and is unique up to some appropriate constraints. We also show how to compute the number of triangulations of a chirotope efficiently, starting from its tree and the (weighted) numbers of triangulations of its parts.
Cite as

Mathilde Bouvel, Valentin Feray, Xavier Goaoc, and Florent Koechlin. A Canonical Tree Decomposition for Chirotopes. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bouvel_et_al:LIPIcs.SoCG.2024.23,
  author =	{Bouvel, Mathilde and Feray, Valentin and Goaoc, Xavier and Koechlin, Florent},
  title =	{{A Canonical Tree Decomposition for Chirotopes}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.23},
  URN =		{urn:nbn:de:0030-drops-199680},
  doi =		{10.4230/LIPIcs.SoCG.2024.23},
  annote =	{Keywords: Order type, modular decomposition, counting triangulations, mutually avoiding point sets, generating functions, rewriting systems}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.30
On Connectivity in Random Graph Models with Limited Dependencies

Authors: Johannes Lengler, Anders Martinsson, Kalina Petrova, Patrick Schnider, Raphael Steiner, Simon Weber, and Emo Welzl

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

Abstract
For any positive edge density p, a random graph in the Erdős-Rényi G_{n,p} model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share endpoints are independent while incident edges may be dependent and ask: what is the minimum probability ρ(n), such that for any distribution 𝒢 (in this model) on graphs with n vertices in which each potential edge has a marginal probability of being present at least ρ(n), a graph drawn from 𝒢 is connected with non-zero probability? As it turns out, the condition "edges that do not share endpoints are independent" needs to be clarified and the answer to the question above is sensitive to the specification. In fact, we formalize this intuitive description into a strict hierarchy of five independence conditions, which we show to have at least three different behaviors for the threshold ρ(n). For each condition, we provide upper and lower bounds for ρ(n). In the strongest condition, the coloring model (which includes, e.g., random geometric graphs), we show that ρ(n) → 2-ϕ ≈ 0.38 for n → ∞, proving a conjecture by Badakhshian, Falgas-Ravry, and Sharifzadeh. This separates the coloring models from the weaker independence conditions we consider, as there we prove that ρ(n) > 0.5-o(n). In stark contrast to the coloring model, for our weakest independence condition - pairwise independence of non-adjacent edges - we show that ρ(n) lies within O(1/n²) of the threshold 1-2/n for completely arbitrary distributions.
Cite as

Johannes Lengler, Anders Martinsson, Kalina Petrova, Patrick Schnider, Raphael Steiner, Simon Weber, and Emo Welzl. On Connectivity in Random Graph Models with Limited Dependencies. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 30:1-30:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lengler_et_al:LIPIcs.APPROX/RANDOM.2023.30,
  author =	{Lengler, Johannes and Martinsson, Anders and Petrova, Kalina and Schnider, Patrick and Steiner, Raphael and Weber, Simon and Welzl, Emo},
  title =	{{On Connectivity in Random Graph Models with Limited Dependencies}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{30:1--30:22},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.30},
  URN =		{urn:nbn:de:0030-drops-188556},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.30},
  annote =	{Keywords: Random Graphs, Independence, Dependency, Connectivity, Threshold, Probabilistic Method}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.22
Improved Algebraic Degeneracy Testing

Authors: Jean Cardinal and Micha Sharir

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
In the classical linear degeneracy testing problem, we are given n real numbers and a k-variate linear polynomial F, for some constant k, and have to determine whether there exist k numbers a_1,…,a_k from the set such that F(a_1,…,a_k) = 0. We consider a generalization of this problem in which F is an arbitrary constant-degree polynomial, we are given k sets of n real numbers, and have to determine whether there exists a k-tuple of numbers, one in each set, on which F vanishes. We give the first improvement over the naïve O^*(n^{k-1}) algorithm for this problem (where the O^*(⋅) notation omits subpolynomial factors). We show that the problem can be solved in time O^*(n^{k - 2 + 4/(k+2)}) for even k and in time O^*(n^{k - 2 + (4k-8)/(k²-5)}) for odd k in the real RAM model of computation. We also prove that for k = 4, the problem can be solved in time O^*(n^2.625) in the algebraic decision tree model, and for k = 5 it can be solved in time O^*(n^3.56) in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for k-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft’s point-line incidence detection problem in any dimension.
Cite as

Jean Cardinal and Micha Sharir. Improved Algebraic Degeneracy Testing. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cardinal_et_al:LIPIcs.SoCG.2023.22,
  author =	{Cardinal, Jean and Sharir, Micha},
  title =	{{Improved Algebraic Degeneracy Testing}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{22:1--22:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.22},
  URN =		{urn:nbn:de:0030-drops-178723},
  doi =		{10.4230/LIPIcs.SoCG.2023.22},
  annote =	{Keywords: Degeneracy testing, k-SUM problem, incidence bounds, Hocroft’s problem, polynomial method, algebraic decision trees}
}
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