25 Search Results for "De Carufel, Jean-Lou"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.20
The Spanning Ratio of the Directed Θ₆-Graph Is 5

Authors: Prosenjit Bose, Jean-Lou De Carufel, John Stuart, and Darryl Hill

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Given a finite set P ⊂ ℝ², the directed Theta-6 graph, denoted Θ₆(P), is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The Θ₆(P)-graph is defined as follows: the plane around each point u ∈ P is partitioned into 6 equiangular cones with apex u, and in each cone, u is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the Θ₆(P)-graph contains an edge from u to v exactly when the interior of ∇_u^v is disjoint from P, where ∇_u^v is the unique equilateral triangle containing u on a corner, v on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the Θ₆(P)-graph is between 4 and 7 in the worst case (Akitaya, Biniaz, and Bose Comput. Geom., 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any Θ_k(P)-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.
Cite as

Prosenjit Bose, Jean-Lou De Carufel, John Stuart, and Darryl Hill. The Spanning Ratio of the Directed Θ₆-Graph Is 5. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bose_et_al:LIPIcs.SoCG.2026.20,
  author =	{Bose, Prosenjit and De Carufel, Jean-Lou and Stuart, John and Hill, Darryl},
  title =	{{The Spanning Ratio of the Directed \Theta₆-Graph Is 5}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{20:1--20:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.20},
  URN =		{urn:nbn:de:0030-drops-258268},
  doi =		{10.4230/LIPIcs.SoCG.2026.20},
  annote =	{Keywords: Geometric Spanners, Theta Graphs, Directed Theta Graphs, Spanning Ratio, Computational Geometry}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.15
Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

Authors: Sujoy Bhore, Sándor Kisfaludi‑Bak, Lazar Milenković, Csaba D. Tóth, Karol Węgrzycki, and Sampson Wong

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
A Euclidean noncrossing Steiner (1+ε)-spanner for a point set P ⊂ ℝ² is a planar straight-line graph that, for any two points a, b ∈ P, contains a path whose length is at most 1+ε times the Euclidean distance between a and b. We construct a Euclidean noncrossing Steiner (1+ε)-spanner with O(n/ε^{3/2}) edges for any set of n points in the plane. This result improves upon the previous best upper bound of O(n/ε⁴) obtained nearly three decades ago. We also establish an almost matching lower bound: There exist n points in the plane for which any Euclidean noncrossing Steiner (1+ε)-spanner has Ω_μ(n/ε^{3/2-μ}) edges for any μ > 0. Our lower bound uses recent generalizations of the Szemerédi-Trotter theorem to disk-tube incidences in geometric measure theory.
Cite as

Sujoy Bhore, Sándor Kisfaludi‑Bak, Lazar Milenković, Csaba D. Tóth, Karol Węgrzycki, and Sampson Wong. Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.15,
  author =	{Bhore, Sujoy and Kisfaludi‑Bak, S\'{a}ndor and Milenkovi\'{c}, Lazar and T\'{o}th, Csaba D. and W\k{e}grzycki, Karol and Wong, Sampson},
  title =	{{Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{15:1--15:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.15},
  URN =		{urn:nbn:de:0030-drops-258210},
  doi =		{10.4230/LIPIcs.SoCG.2026.15},
  annote =	{Keywords: geometric network design, spanners, crossing number, incidences}
}
Document
DOI: 10.4230/LIPIcs.FUN.2026.30
Replacing Cops with Zombies

Authors: Fengyi Liu and Avery Miller

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract
In the Cops and Robbers game, two players take turns to move their pieces in a given graph, and the goal of the Cop player is to have any of their pieces capture the piece controlled by the Robber player. In the Zombies and Survivors variant, the Zombie player is trying to catch the piece controlled by the Survivor player, but the pursuers are more restricted: the zombie pieces must always move, and must do so toward the survivor along a shortest path. Essentially, the zombies move in a 'brainless' way, which severely limits the design of algorithmic strategies. In a practical setting where the pursuers are robots or drones, the zombie strategies can be carried out by simpler/cheaper devices. This motivates the question: when can cops be replaced by zombies? In contrast to previous work that highlights graph classes where cops are significantly more useful when trying to catch the evader, we examine some situations where the simpler restricted movements imposed on zombies can be sufficient.
Cite as

Fengyi Liu and Avery Miller. Replacing Cops with Zombies. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 30:1-30:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{liu_et_al:LIPIcs.FUN.2026.30,
  author =	{Liu, Fengyi and Miller, Avery},
  title =	{{Replacing Cops with Zombies}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{30:1--30:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.30},
  URN =		{urn:nbn:de:0030-drops-257492},
  doi =		{10.4230/LIPIcs.FUN.2026.30},
  annote =	{Keywords: Pursuit-Evasion Games, Grid Graphs, Cop Number, Zombie Number, Throttling Number}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.16
A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth

Authors: Sergio Cabello, Alexander Dobler, Gašper Fijavž, Thekla Hamm, and Mirko H. Wagner

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type characterizes the subgraph induced by the four vertices of the crossing edges. Each of the 63 possible nonempty subsets 𝒮 of crossing types gives a recognition problem: does a given graph admit an 𝒮-restricted drawing, that is, a 1-planar drawing where the crossing type of each crossing is in 𝒮? We show that there is a set 𝒮_bad with three crossing types and the following properties: - If 𝒮 contains no crossing type from 𝒮_bad, then the recognition of graphs that admit an 𝒮-restricted drawing is fixed-parameter tractable with respect to the treewidth of the input graph. - If 𝒮 contains any crossing type from 𝒮_bad, then it is NP-hard to decide whether a graph has an 𝒮-restricted drawing, even when considering graphs of constant pathwidth. We also extend this characterization of crossing types to 1-planar straight-line drawings and show the same complexity behaviour parameterized by treewidth.
Cite as

Sergio Cabello, Alexander Dobler, Gašper Fijavž, Thekla Hamm, and Mirko H. Wagner. A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cabello_et_al:LIPIcs.ISAAC.2025.16,
  author =	{Cabello, Sergio and Dobler, Alexander and Fijav\v{z}, Ga\v{s}per and Hamm, Thekla and Wagner, Mirko H.},
  title =	{{A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{16:1--16:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.16},
  URN =		{urn:nbn:de:0030-drops-249248},
  doi =		{10.4230/LIPIcs.ISAAC.2025.16},
  annote =	{Keywords: 1-planar, crossing type, treewidth, pathwidth}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.75
An Improved Bound for Plane Covering Paths

Authors: Hugo A. Akitaya, Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Cyril Gavoille, John Iacono, Linda Kleist, Michiel Smid, Diane Souvaine, and Leonidas Theocharous

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
A covering path for a finite set P of points in the plane is a polygonal path such that every point of P lies on a segment of the path. The vertices of the path need not be at points of P. A covering path is plane if its segments do not cross each other. Let π(n) be the minimum number such that every set of n points in the plane admits a plane covering path with at most π(n) segments. We prove that π(n) ≤ ⌈6n/7⌉. This improves the previous best-known upper bound of ⌈21n/22⌉, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple O(n log n)-time algorithm for computing a plane covering path.
Cite as

Hugo A. Akitaya, Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Cyril Gavoille, John Iacono, Linda Kleist, Michiel Smid, Diane Souvaine, and Leonidas Theocharous. An Improved Bound for Plane Covering Paths. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 75:1-75:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{a.akitaya_et_al:LIPIcs.ESA.2025.75,
  author =	{A. Akitaya, Hugo and Aloupis, Greg and Biniaz, Ahmad and Bose, Prosenjit and De Carufel, Jean-Lou and Gavoille, Cyril and Iacono, John and Kleist, Linda and Smid, Michiel and Souvaine, Diane and Theocharous, Leonidas},
  title =	{{An Improved Bound for Plane Covering Paths}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{75:1--75:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.75},
  URN =		{urn:nbn:de:0030-drops-245432},
  doi =		{10.4230/LIPIcs.ESA.2025.75},
  annote =	{Keywords: Covering Path, Upper Bound, Simple Algorithm}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.8
Tight Bounds on the Number of Closest Pairs in Vertical Slabs

Authors: Ahmad Biniaz, Prosenjit Bose, Chaeyoon Chung, Jean-Lou De Carufel, John Iacono, Anil Maheshwari, Saeed Odak, Michiel Smid, and Csaba D. Tóth

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

Abstract
Let S be a set of n points in ℝ^d, where d ≥ 2 is a constant, and let H₁,H₂,…,H_{m+1} be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly n/m points of S are between any two successive hyperplanes. Let |A(S,m)| be the number of different closest pairs in the {(m+1) choose 2} vertical slabs that are bounded by H_i and H_j, over all 1 ≤ i < j ≤ m+1. We prove tight bounds for the largest possible value of |A(S,m)|, over all point sets of size n, and for all values of 1 ≤ m ≤ n. As a result of these bounds, we obtain, for any constant ε > 0, a data structure of size O(n), such that for any vertical query slab Q, the closest pair in the set Q ∩ S can be reported in O(n^{1/2+ε}) time. Prior to this work, no linear space data structure with sublinear query time was known.
Cite as

Ahmad Biniaz, Prosenjit Bose, Chaeyoon Chung, Jean-Lou De Carufel, John Iacono, Anil Maheshwari, Saeed Odak, Michiel Smid, and Csaba D. Tóth. Tight Bounds on the Number of Closest Pairs in Vertical Slabs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{biniaz_et_al:LIPIcs.WADS.2025.8,
  author =	{Biniaz, Ahmad and Bose, Prosenjit and Chung, Chaeyoon and De Carufel, Jean-Lou and Iacono, John and Maheshwari, Anil and Odak, Saeed and Smid, Michiel and T\'{o}th, Csaba D.},
  title =	{{Tight Bounds on the Number of Closest Pairs in Vertical Slabs}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{8:1--8:14},
  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.2025.8},
  URN =		{urn:nbn:de:0030-drops-242391},
  doi =		{10.4230/LIPIcs.WADS.2025.8},
  annote =	{Keywords: closest pair, vertical slab, data structure}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.9
Online Routing in Directed Yao₄^∞ Graphs

Authors: Prosenjit Bose, Jean-Lou De Carufel, and John Stuart

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

Abstract
The x⃑{Yao₄^∞} and x⃑{Yao₄} graphs are two families of directed geometric graphs whose vertices are points in the plane, and each vertex has up to four outgoing edges. Consider a horizontal and a vertical line through each vertex v, defining four quadrants around v. Then v has an outgoing edge to the closest vertex in each of its four quadrants. When the distance is measured using the Euclidean norm, the resulting graph is the x⃑{Yao₄} graph, whereas with the L_∞-norm, we obtain the x⃑{Yao^{∞}₄} graph, which is a sub-graph of the well-known L_∞-Delaunay graph. In this paper, we provide a local routing algorithm with routing ratio at most 85.22 for x⃑{Yao^{∞}₄} graphs. Prior to this work, no constant spanning or routing ratios for x⃑{Yao₄^∞} graphs were previously known. Now, x⃑{Yao₄^∞} graphs are the sparsest family of directed planar graphs supporting a competitive local routing strategy. Furthermore, we show that no local routing algorithm for x⃑{Yao₄^∞} graphs can have a routing ratio lower than 7+√2≈ 8.41. Moreover, we prove that the spanning ratio is at least 5+√2≈ 6.41 in the worst case. The techniques we develop in this paper also allow us to prove lower bounds of 7-√3+√{5-2√3}≈ 6.51 and 7+√2 for the spanning and routing ratios of x⃑{Yao₄}, respectively.
Cite as

Prosenjit Bose, Jean-Lou De Carufel, and John Stuart. Online Routing in Directed Yao₄^∞ Graphs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bose_et_al:LIPIcs.WADS.2025.9,
  author =	{Bose, Prosenjit and De Carufel, Jean-Lou and Stuart, John},
  title =	{{Online Routing in Directed Yao₄^∞ Graphs}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{9:1--9:22},
  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.2025.9},
  URN =		{urn:nbn:de:0030-drops-242404},
  doi =		{10.4230/LIPIcs.WADS.2025.9},
  annote =	{Keywords: Geometric Spanners, Yao Graphs, Local Routing Algorithms}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.33
Spanner for the 0/1/∞ Weighted Region Problem

Authors: Joachim Gudmundsson, Zijin Huang, André van Renssen, and Sampson Wong

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

Abstract
We consider the problem of computing an approximate weighted shortest path in a weighted planar subdivision, with weights assigned from the set {0, 1, ∞}. The subdivision includes zero-cost regions (0-regions) with weight 0 and obstacles with weight ∞, all embedded in a plane with weight 1. In a polygonal domain, where the 0-regions and obstacles are non-overlapping polygons (not necessarily convex) with in total N vertices, we present an algorithm that computes a (1 + ε)-approximate spanner of the input vertices in expected Õ(N/ε³) time, for 0 < ε < 1. Using our spanner, we can compute a (1 + ε)-approximate weighted shortest path between any two points (not necessarily vertices) in Õ(N/ε³) time. Furthermore, we prove that our results more generally apply to non-polygonal convex regions. Using this generalisation, one can approximate the weak partial Fréchet similarity [Buchin et al., 2009] between two polygonal curves in expected Õ(n²/ε²) time, where n is the total number of vertices of the input curves.
Cite as

Joachim Gudmundsson, Zijin Huang, André van Renssen, and Sampson Wong. Spanner for the 0/1/∞ Weighted Region Problem. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gudmundsson_et_al:LIPIcs.WADS.2025.33,
  author =	{Gudmundsson, Joachim and Huang, Zijin and van Renssen, Andr\'{e} and Wong, Sampson},
  title =	{{Spanner for the 0/1/∞ Weighted Region Problem}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{33:1--33:15},
  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.2025.33},
  URN =		{urn:nbn:de:0030-drops-242644},
  doi =		{10.4230/LIPIcs.WADS.2025.33},
  annote =	{Keywords: weighted region problem, approximate shortest path, spanner}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.34
Link Diameter, Radius and 2-Point Link Distance Queries in Polygonal Domains

Authors: Mart Hagedoorn and Valentin Polishchuk

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

Abstract
We show how to preprocess a polygonal domain with holes so that the link distance (the number of links in a minimum-link path) between two query points in the domain can be reported efficiently. Using our data structures, the link diameter of the domain (i.e., the maximum number of links that may be required in a minimum-link path between two points in the domain) as well as the link center and radius of the domain (i.e., the point minimizing the maximum link distance to the furthest point in the domain and this maximum link distance) can be found in polynomial time. We also give a simpler algorithm for finding the link diameter, not using the link distance query structures. Answering 2-point link distance queries and computing the link diameter/radius/center in polygonal domains have been open questions since these problems were studied for simple polygons in the 90’s.
Cite as

Mart Hagedoorn and Valentin Polishchuk. Link Diameter, Radius and 2-Point Link Distance Queries in Polygonal Domains. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hagedoorn_et_al:LIPIcs.WADS.2025.34,
  author =	{Hagedoorn, Mart and Polishchuk, Valentin},
  title =	{{Link Diameter, Radius and 2-Point Link Distance Queries in Polygonal Domains}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{34:1--34:15},
  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.2025.34},
  URN =		{urn:nbn:de:0030-drops-242659},
  doi =		{10.4230/LIPIcs.WADS.2025.34},
  annote =	{Keywords: Minimum-link paths, link distance, diameter, center, radius, 2-point distance queries}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.75
Temporal Graph Realization with Bounded Stretch

Authors: George B. Mertzios, Hendrik Molter, Nils Morawietz, and Paul G. Spirakis

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
A periodic temporal graph, in its simplest form, is a graph in which every edge appears exactly once in the first Δ time steps, and then it reappears recurrently every Δ time steps, where Δ is a given period length. This model offers a natural abstraction of transportation networks where each transportation link connects two destinations periodically. From a network design perspective, a crucial task is to assign the time-labels on the edges in a way that optimizes some criterion. In this paper we introduce a very natural optimality criterion that captures how the temporal distances of all vertex pairs are "stretched", compared to their physical distances, i.e. their distances in the underlying static (non-temporal) graph. Given a static graph G, the task is to assign to each edge one time-label between 1 and Δ such that, in the resulting periodic temporal graph with period Δ, the duration of the fastest temporal path from any vertex u to any other vertex v is at most α times the distance between u and v in G. Here, the value of α measures how much the shortest paths are allowed to be stretched once we assign the periodic time-labels. Our results span three different directions: First, we provide a series of approximation and NP-hardness results. Second, we provide approximation and fixed-parameter algorithms. Among them, we provide a simple polynomial-time algorithm (the radius-algorithm) which always guarantees an approximation strictly smaller than Δ, and which also computes the optimum stretch in some cases. Third, we consider a parameterized local search extension of the problem where we are given the temporal labeling of the graph, but we are allowed to change the time-labels of at most k edges; for this problem we prove that it is W[2]-hard but admits an XP algorithm with respect to k.
Cite as

George B. Mertzios, Hendrik Molter, Nils Morawietz, and Paul G. Spirakis. Temporal Graph Realization with Bounded Stretch. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 75:1-75:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mertzios_et_al:LIPIcs.MFCS.2025.75,
  author =	{Mertzios, George B. and Molter, Hendrik and Morawietz, Nils and Spirakis, Paul G.},
  title =	{{Temporal Graph Realization with Bounded Stretch}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{75:1--75:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.75},
  URN =		{urn:nbn:de:0030-drops-241829},
  doi =		{10.4230/LIPIcs.MFCS.2025.75},
  annote =	{Keywords: Temporal graph, periodic temporal labeling, fastest temporal path, graph realization, temporal connectivity, stretch}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.27
Cops and Robbers for Graphs on Surfaces with Crossings

Authors: Prosenjit Bose, Pat Morin, and Karthik Murali

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Cops and Robbers is a game played on a graph where a set of cops attempt to capture a single robber. The game proceeds in rounds, where each round first consists of the cops' turn, followed by the robber’s turn. In the first round, the cops place themselves on a subset of vertices, after which the robber chooses a vertex to place himself. From the next round onwards, in the cops' turn, every cop can choose to either stay on the same vertex or move to an adjacent vertex, and likewise the robber in his turn. The robber is considered to be captured if, at any point in time, there is some cop on the same vertex as the robber. The cops win if they can capture the robber within a finite number of rounds; else the robber wins. A natural question in this game concerns the cop-number of a graph - the minimum number of cops needed to capture a robber. It has long been known that graphs embeddable (without crossings) on surfaces of bounded genus have bounded cop-number. In contrast, it was shown recently that the class of 1-planar graphs - graphs that can be drawn on the plane with at most one crossing per edge - does not have bounded cop-number. This paper initiates an investigation into how the distance between crossing pairs of edges influences a graph’s cop number. In particular, we look at Distance d Cops and Robbers, a variant of the classical game, where the robber is considered to be captured if there is a cop within distance d of the robber. Let c_d(G) denote the minimum number of cops required in the graph G to capture a robber within distance d. We look at various classes of graphs, such as 1-plane graphs, k-plane graphs (graphs where each edge is crossed at most k times), and even general graph drawings, and show that if every crossing pair of edges can be connected by a path of small length, then c_d(G) is bounded, for small values of d. For example, we show that if a graph G admits a drawing in which every pair of crossing edges is contained in a path of length at most 3, then c₄(G) ≤ 21. And if the drawing permits a stronger assumption that the endpoints of every crossing induce the complete graph K₄, then c₃(G) ≤ 9. The tools and techniques that we develop in this paper are sufficiently general, enabling us to examine graphs drawn not only on the sphere but also on orientable and non-orientable surfaces.
Cite as

Prosenjit Bose, Pat Morin, and Karthik Murali. Cops and Robbers for Graphs on Surfaces with Crossings. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bose_et_al:LIPIcs.MFCS.2025.27,
  author =	{Bose, Prosenjit and Morin, Pat and Murali, Karthik},
  title =	{{Cops and Robbers for Graphs on Surfaces with Crossings}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{27:1--27:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.27},
  URN =		{urn:nbn:de:0030-drops-241349},
  doi =		{10.4230/LIPIcs.MFCS.2025.27},
  annote =	{Keywords: Cops and Robbers, Crossings, 1-Planar, Surfaces}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2025.7
Temporal Explorability Games

Authors: Pete Austin, Sougata Bose, Nicolas Mazzocchi, and Patrick Totzke

Published in: LIPIcs, Volume 348, 36th International Conference on Concurrency Theory (CONCUR 2025)

Abstract
Temporal graphs extend ordinary graphs with discrete time that affects the availability of edges. We consider solving games played on temporal graphs where one player aims to explore the graph, i.e., visit all vertices. The complexity depends majorly on two factors: the presence of an adversary and how edge availability is specified. We demonstrate that on static graphs, where edges are always available, solving explorability games is just as hard as solving reachability games. In contrast, on temporal graphs, the complexity of explorability coincides with generalized reachability (NP-complete for one-player and PSPACE-complete for two player games). We show that if temporal graphs are given symbolically, even one-player reachability (and thus explorability and generalized reachability) games are PSPACE-hard. For one player, all these are also solvable in PSPACE and for two players, they are in PSPACE, EXP and EXP, respectively.
Cite as

Pete Austin, Sougata Bose, Nicolas Mazzocchi, and Patrick Totzke. Temporal Explorability Games. In 36th International Conference on Concurrency Theory (CONCUR 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 348, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{austin_et_al:LIPIcs.CONCUR.2025.7,
  author =	{Austin, Pete and Bose, Sougata and Mazzocchi, Nicolas and Totzke, Patrick},
  title =	{{Temporal Explorability Games}},
  booktitle =	{36th International Conference on Concurrency Theory (CONCUR 2025)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-389-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{348},
  editor =	{Bouyer, Patricia and van de Pol, Jaco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2025.7},
  URN =		{urn:nbn:de:0030-drops-239575},
  doi =		{10.4230/LIPIcs.CONCUR.2025.7},
  annote =	{Keywords: Temporal Graphs, Explorability, Reachability, Games}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.19
Polychromatic Coloring of Tuples in Hypergraphs

Authors: Ahmad Biniaz, Jean-Lou De Carufel, Anil Maheshwari, Michiel Smid, Shakhar Smorodinsky, and Miloš Stojaković

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

Abstract
A hypergraph H consists of a set V of vertices and a set E of hyperedges that are subsets of V. A t-tuple of H is a subset of t vertices of V. A t-tuple k-coloring of H is a mapping of its t-tuples into k colors. A coloring is called (t,k,f)-polychromatic if each hyperedge of E that has at least f vertices contains tuples of all the k colors. Let f_H(t,k) be the minimum f such that H has a (t,k,f)-polychromatic coloring. For a family of hypergraphs ℋ let f_H(t,k) be the maximum f_H(t,k) over all hypergraphs H in H. Determining f_H(t,k) has been an active research direction in recent years. This is challenging even for t = 1. We present several new results in this direction for t ≥ 2. - Let H be the family of hypergraphs H that is obtained by taking any set P of points in ℝ², setting V: = P and E: = {d ∩ P: d is a disk in ℝ²}. We prove that f_ H(2,k) ≤ 3.7^k, that is, the pairs of points (2-tuples) can be k-colored such that any disk containing at least 3.7^k points has pairs of all colors. We generalize this result to points and balls in higher dimensions. - For the family H of hypergraphs that are defined by grid vertices and axis-parallel rectangles in the plane, we show that f_H(2,k) ≤ √{ck ln k} for some constant c. We then generalize this to higher dimensions, to other shapes, and to tuples of larger size. - For the family H of shrinkable hypergraphs of VC-dimension at most d we prove that f_ H(d+1,k) ≤ c^k for some constant c = c(d). Towards this bound, we obtain a result of independent interest: Every hypergraph with n vertices and with VC-dimension at most d has a (d+1)-tuple T of depth at least n/c, i.e., any hyperedge that contains T also contains n/c other vertices. - For the relationship between t-tuple coloring and vertex coloring in any hypergraph H we establish the inequality 1/e⋅ tk^{1/t} ≤ f_H(t,k) ≤ f_H(1,tk^{1/t}). For the special case of k = 2, referred to as the bichromatic coloring, we prove that t+1 ≤ f_H(t,2) ≤ max{f_H(1,2), t+1}; this improves upon the previous best known upper bound. - We study the relationship between tuple coloring and epsilon nets. In particular we show that if f_H(1,k) = O(k) for a hypergraph H with n vertices, then for any 0 < ε < 1 the t-tuples of H can be partitioned into Ω((εn/t)^t) ε-t-nets. This bound is tight when t is a constant.
Cite as

Ahmad Biniaz, Jean-Lou De Carufel, Anil Maheshwari, Michiel Smid, Shakhar Smorodinsky, and Miloš Stojaković. Polychromatic Coloring of Tuples in Hypergraphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{biniaz_et_al:LIPIcs.SoCG.2025.19,
  author =	{Biniaz, Ahmad and De Carufel, Jean-Lou and Maheshwari, Anil and Smid, Michiel and Smorodinsky, Shakhar and Stojakovi\'{c}, Milo\v{s}},
  title =	{{Polychromatic Coloring of Tuples in Hypergraphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{19:1--19:17},
  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.2025.19},
  URN =		{urn:nbn:de:0030-drops-231718},
  doi =		{10.4230/LIPIcs.SoCG.2025.19},
  annote =	{Keywords: Hypergraph Coloring, Polychromatic Coloring, Geometric Hypergraphs, Cover Decomposable Hypergraphs, Epsilon Nets}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.48
Exact Algorithms for Minimum Dilation Triangulation

Authors: Sándor P. Fekete, Phillip Keldenich, and Michael Perk

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

Abstract
We provide a spectrum of new theoretical insights and practical results for finding a Minimum Dilation Triangulation (MDT), a natural geometric optimization problem of considerable previous attention: Given a set P of n points in the plane, find a triangulation T, such that a shortest Euclidean path in T between any pair of points increases by the smallest possible factor compared to their straight-line distance. No polynomial-time algorithm is known for the problem; moreover, evaluating the objective function involves computing the sum of (possibly many) square roots. On the other hand, the problem is not known to be NP-hard. (1) We provide practically robust methods and implementations for computing an MDT for benchmark sets with up to 30,000 points in reasonable time on commodity hardware, based on new geometric insights into the structure of optimal edge sets. Previous methods only achieved results for up to 200 points, so we extend the range of optimally solvable instances by a factor of 150. (2) We develop scalable techniques for accurately evaluating many shortest-path queries that arise as large-scale sums of square roots, allowing us to certify exact optimal solutions, with previous work relying on (possibly inaccurate) floating-point computations. (3) We resolve an open problem by establishing a lower bound of 1.44116 on the dilation of the regular 84-gon (and thus for arbitrary point sets), improving the previous worst-case lower bound of 1.4308 and greatly reducing the remaining gap to the upper bound of 1.4482 from the literature. In the process, we provide optimal solutions for regular n-gons up to n = 100.
Cite as

Sándor P. Fekete, Phillip Keldenich, and Michael Perk. Exact Algorithms for Minimum Dilation Triangulation. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fekete_et_al:LIPIcs.SoCG.2025.48,
  author =	{Fekete, S\'{a}ndor P. and Keldenich, Phillip and Perk, Michael},
  title =	{{Exact Algorithms for Minimum Dilation Triangulation}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{48:1--48:18},
  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.2025.48},
  URN =		{urn:nbn:de:0030-drops-232006},
  doi =		{10.4230/LIPIcs.SoCG.2025.48},
  annote =	{Keywords: dilation, minimum dilation triangulation, exact algorithms, algorithm engineering, experimental evaluation}
}
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.
Cite as

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}
}
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