Volume
LIPIcs, Volume 332
41st International Symposium on Computational Geometry (SoCG 2025)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Computational Geometry (SoCG)
Event
SoCG 2025, June 23-27, 2025, Kanazawa, JapanEditors
Publication Details
- published at: 2025-06-20
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-370-6
- DBLP: db/conf/compgeom/compgeom2025
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Complete Volume
LIPIcs, Volume 332, SoCG 2025, Complete Volume
Abstract
LIPIcs, Volume 332, SoCG 2025, Complete Volume
Cite as
41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 1-1346, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@Proceedings{aichholzer_et_al:LIPIcs.SoCG.2025,
title = {{LIPIcs, Volume 332, SoCG 2025, Complete Volume}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {1--1346},
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},
URN = {urn:nbn:de:0030-drops-235578},
doi = {10.4230/LIPIcs.SoCG.2025},
annote = {Keywords: LIPIcs, Volume 332, SoCG 2025, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 0:i-0:xxii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2025.0,
author = {Aichholzer, Oswin and Wang, Haitao},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {0:i--0:xxii},
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.2025.0},
URN = {urn:nbn:de:0030-drops-235566},
doi = {10.4230/LIPIcs.SoCG.2025.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Reconfiguration of Unit Squares and Disks: PSPACE-Hardness in Simple Settings
Abstract
We study well-known reconfiguration problems. Given a start and a target configuration of geometric objects in a polygon, we wonder whether we can move the objects from the start configuration to the target configuration while avoiding collisions between the objects and staying within the polygon. Problems of this type have been considered since the early 80s by roboticists and computational geometers. In this paper, we study some of the simplest possible variants where the objects are labeled or unlabeled unit squares or unit disks. In unlabeled reconfiguration, the objects are identical, so that any object is allowed to end at any of the targets positions. In the labeled variant, each object has a designated target position. The results for the labeled variants are direct consequences from our insights on the unlabeled versions.
We show that it is PSPACE-hard to decide whether there exists a reconfiguration of (unlabeled/labeled) unit squares even in a simple polygon. Previously, it was only known to be PSPACE-hard in a polygon with holes for both the unlabeled and labeled version [Solovey and Halperin, Int. J. Robotics Res. 2016]. Our proof is based on a result of independent interest, namely that reconfiguration between two satisfying assignments of a formula of Monotone-Planar-3-Sat is also PSPACE-complete. The reduction from reconfiguration of Monotone-Planar-3-Sat to reconfiguration of unit squares extends techniques recently developed to show NP-hardness of packing unit squares in a simple polygon [Abrahamsen and Stade, FOCS 2024]. We also show PSPACE-hardness of reconfiguration of (unlabeled/labeled) unit disks in a polygon with holes. Previously, it was known that unlabeled reconfiguration of disks of two different sizes was PSPACE-hard [Brocken, van der Heijden, Kostitsyna, Lo-Wong and Surtel, FUN 2021].
Cite as
Mikkel Abrahamsen, Kevin Buchin, Maike Buchin, Linda Kleist, Maarten Löffler, Lena Schlipf, André Schulz, and Jack Stade. Reconfiguration of Unit Squares and Disks: PSPACE-Hardness in Simple Settings. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2025.1,
author = {Abrahamsen, Mikkel and Buchin, Kevin and Buchin, Maike and Kleist, Linda and L\"{o}ffler, Maarten and Schlipf, Lena and Schulz, Andr\'{e} and Stade, Jack},
title = {{Reconfiguration of Unit Squares and Disks: PSPACE-Hardness in Simple Settings}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {1:1--1:18},
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.2025.1},
URN = {urn:nbn:de:0030-drops-231539},
doi = {10.4230/LIPIcs.SoCG.2025.1},
annote = {Keywords: reconfiguration, unit square, unit disk, unlabeled, labeled, simple polygon, polygon}
}
Document
The Maximum Number of Digons Formed by Pairwise Intersecting Pseudocircles
Abstract
In 1972, Branko Grünbaum conjectured that any nontrivial arrangement of n > 2 pairwise intersecting pseudocircles in the plane can have at most 2n-2 digons (regions enclosed by exactly two pseudoarcs), with the bound being tight. While this conjecture has been confirmed for cylindrical arrangements of pseudocircles and more recently for geometric circles, we extend these results to any simple arrangement of pairwise intersecting pseudocircles.
Cite as
Eyal Ackerman, Gábor Damásdi, Balázs Keszegh, Rom Pinchasi, and Rebeka Raffay. The Maximum Number of Digons Formed by Pairwise Intersecting Pseudocircles. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ackerman_et_al:LIPIcs.SoCG.2025.2,
author = {Ackerman, Eyal and Dam\'{a}sdi, G\'{a}bor and Keszegh, Bal\'{a}zs and Pinchasi, Rom and Raffay, Rebeka},
title = {{The Maximum Number of Digons Formed by Pairwise Intersecting Pseudocircles}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {2:1--2:14},
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.2025.2},
URN = {urn:nbn:de:0030-drops-231548},
doi = {10.4230/LIPIcs.SoCG.2025.2},
annote = {Keywords: pairwise intersecting arrangement, arrangement of pseudocircles, counting digons, tangencies, Gr\"{u}nbaum’s conjecture}
}
Document
Convexity Helps Iterated Search in 3D
Abstract
Inspired by the classical fractional cascading technique [Bernard Chazelle and Leonidas J. Guibas, 1986; Bernard Chazelle and Leonidas J. Guibas, 1986], we introduce new techniques to speed up the following type of iterated search in 3D: The input is a graph 𝐆 with bounded degree together with a set H_v of 3D hyperplanes associated with every vertex of v of 𝐆. The goal is to store the input such that given a query point q ∈ ℝ³ and a connected subgraph 𝐇 ⊂ 𝐆, we can decide if q is below or above the lower envelope of H_v for every v ∈ 𝐇. We show that using linear space, it is possible to answer queries in roughly O(log n + |𝐇|√{log n}) time which improves trivial bound of O(|𝐇|log n) obtained by using planar point location data structures. Our data structure can in fact answer more general queries (it combines with shallow cuttings) and it even works when 𝐇 is given one vertex at a time. We show that this has a number of new applications and in particular, we give improved solutions to a set of natural data structure problems that up to our knowledge had not seen any improvements.
We believe this is a very surprising result because obtaining similar results for the planar point location problem was known to be impossible [Chazelle and Liu, 2004].
Cite as
Peyman Afshani, Yakov Nekrich, and Frank Staals. Convexity Helps Iterated Search in 3D. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{afshani_et_al:LIPIcs.SoCG.2025.3,
author = {Afshani, Peyman and Nekrich, Yakov and Staals, Frank},
title = {{Convexity Helps Iterated Search in 3D}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {3:1--3:14},
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.2025.3},
URN = {urn:nbn:de:0030-drops-231558},
doi = {10.4230/LIPIcs.SoCG.2025.3},
annote = {Keywords: Data structures, range searching}
}
Document
A Subquadratic Algorithm for Computing the L₁-Distance Between Two Terrains
Abstract
We study the problem of computing the L₁-distance between two piecewise-linear bivariate functions f and g, defined over a bounded polygonal domain 𝕄 ⊂ ℝ², that is, computing the quantity ‖f-g‖₁ = ∫_𝕄 |f(x,y)-g(x,y)| dx dy. If f and g are defined by linear interpolation over triangulations 𝐓_f and 𝐓_g, respectively, of 𝕄 with a total of n triangles, we show that ‖f-g‖₁ can be computed in Õ(n^α) time, where α = max{(ω+1)/2, 8/5}, ω is the matrix multiplication exponent, and Õ notation hides factors of the form n^ε for any ε > 0. This bound holds for the currently best known value of ω, which is approximately 2.37. More generally, if the complexity of the overlay of 𝐓_f and 𝐓_g is κ, then the runtime of our algorithm is Õ(κ^{α-1}n^{2-α}).
Cite as
Pankaj K. Agarwal, Boris Aronov, Olivier Devillers, Christian Knauer, and Guillaume Moroz. A Subquadratic Algorithm for Computing the L₁-Distance Between Two Terrains. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2025.4,
author = {Agarwal, Pankaj K. and Aronov, Boris and Devillers, Olivier and Knauer, Christian and Moroz, Guillaume},
title = {{A Subquadratic Algorithm for Computing the L₁-Distance Between Two Terrains}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {4:1--4:17},
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.2025.4},
URN = {urn:nbn:de:0030-drops-231561},
doi = {10.4230/LIPIcs.SoCG.2025.4},
annote = {Keywords: Terrain similarity, volume computation, polynomial interpolation, geometric cuttings, bivariate multipoint evaluation}
}
Document
Optimal Motion Planning for Two Square Robots in a Rectilinear Environment
Abstract
Let W ⊂ ℝ² be a rectilinear polygonal environment (that is, a rectilinear polygon potentially with holes) with a total of n vertices, and let A,B be two robots, each modeled as an axis-aligned unit square, that can move rectilinearly inside W. The goal is to compute an optimal collision-free motion plan π for A and B between a given pair of source and target configurations. We study two variants of this problem and obtain the following results.
- Min-Sum: Here the goal is to compute a motion plan that minimizes the sum of the lengths of the paths of the robots. We present an O(n⁴log n)-time algorithm for computing an optimal solution to the min-sum problem. This is the first polynomial-time algorithm to compute an optimal, collision-free motion of two robots amid obstacles in a planar polygonal environment.
- Min-Makespan: Here the robots can move with at most unit speed, and the goal is to compute a motion plan that minimizes the maximum time taken by a robot to reach its target location. We prove that the min-makespan variant is NP-hard.
Cite as
Pankaj K. Agarwal, Mark de Berg, Benjamin Holmgren, Alex Steiger, and Martijn Struijs. Optimal Motion Planning for Two Square Robots in a Rectilinear Environment. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2025.5,
author = {Agarwal, Pankaj K. and de Berg, Mark and Holmgren, Benjamin and Steiger, Alex and Struijs, Martijn},
title = {{Optimal Motion Planning for Two Square Robots in a Rectilinear Environment}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {5:1--5:17},
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.2025.5},
URN = {urn:nbn:de:0030-drops-231577},
doi = {10.4230/LIPIcs.SoCG.2025.5},
annote = {Keywords: Computational geometry, motion planning, multiple robots, rectilinear paths}
}
Document
A Sparse Multicover Bifiltration of Linear Size
Abstract
The k-cover of a point cloud X of ℝ^d at radius r is the set of all those points within distance r of at least k points of X. By varying r and k we obtain a two-parameter filtration known as the multicover bifiltration. This bifiltration has received attention recently due to being choice-free and robust to outliers. However, it is hard to compute: the smallest known equivalent simplicial bifiltration has O(|X|^{d+1}) simplices. In this paper we introduce a (1+ε)-approximation of the multicover bifiltration of linear size O(|X|), for fixed d and ε. The methods also apply to the subdivision Rips bifiltration on metric spaces of bounded doubling dimension yielding analogous results.
Cite as
Ángel Javier Alonso. A Sparse Multicover Bifiltration of Linear Size. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{alonso:LIPIcs.SoCG.2025.6,
author = {Alonso, \'{A}ngel Javier},
title = {{A Sparse Multicover Bifiltration of Linear Size}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {6:1--6:18},
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.2025.6},
URN = {urn:nbn:de:0030-drops-231587},
doi = {10.4230/LIPIcs.SoCG.2025.6},
annote = {Keywords: Multicover, Approximation, Sparsification, Multiparameter persistence}
}
Document
Single-Source Shortest Path Problem in Weighted Disk Graphs
Abstract
In this paper, we present efficient algorithms for the single-source shortest path problem in weighted disk graphs. A disk graph is the intersection graph of a family of disks in the plane. Here, the weight of an edge is defined as the Euclidean distance between the centers of the disks corresponding to the endpoints of the edge. Given a family of n disks in the plane whose radii lie in [1,Ψ] and a source disk, we can compute a shortest path tree from a source vertex in the weighted disk graph in O(nlog² n log Ψ) time. Moreover, in the case that the radii of disks are arbitrarily large, we can compute a shortest path tree from a source vertex in the weighted disk graph in O(nlog⁴ n) time. This improves the best-known algorithm running in O(nlog⁶ n) time presented in ESA'23.
Cite as
Shinwoo An, Eunjin Oh, and Jie Xue. Single-Source Shortest Path Problem in Weighted Disk Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{an_et_al:LIPIcs.SoCG.2025.7,
author = {An, Shinwoo and Oh, Eunjin and Xue, Jie},
title = {{Single-Source Shortest Path Problem in Weighted Disk Graphs}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {7:1--7:15},
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.2025.7},
URN = {urn:nbn:de:0030-drops-231594},
doi = {10.4230/LIPIcs.SoCG.2025.7},
annote = {Keywords: Disk graphs, shortest path problem, compressed quadtrees}
}
Document
Rapid Mixing of the Flip Chain over Non-Crossing Spanning Trees
Abstract
We show that the flip chain for non-crossing spanning trees of n+1 points in convex position mixes in time O(n⁸log n). We use connections between Fuss-Catalan structures to construct a comparison argument with a chain similar to Wilson’s lattice path chain (Wilson 2004).
Cite as
Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, Mark Jerrum, and Jiaheng Wang. Rapid Mixing of the Flip Chain over Non-Crossing Spanning Trees. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{anand_et_al:LIPIcs.SoCG.2025.8,
author = {Anand, Konrad and Feng, Weiming and Freifeld, Graham and Guo, Heng and Jerrum, Mark and Wang, Jiaheng},
title = {{Rapid Mixing of the Flip Chain over Non-Crossing Spanning Trees}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {8:1--8:18},
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.2025.8},
URN = {urn:nbn:de:0030-drops-231607},
doi = {10.4230/LIPIcs.SoCG.2025.8},
annote = {Keywords: non-crossing spanning trees, Markov chain, mixing time}
}
Document
Geometric Realizations of Dichotomous Ordinal Graphs
Abstract
A dichotomous ordinal graph consists of an undirected graph with a partition of the edges into short and long edges. A geometric realization of a dichotomous ordinal graph G in a metric space X is a drawing of G in X in which every long edge is strictly longer than every short edge. We call a graph G pandichotomous in X if G admits a geometric realization in X for every partition of its edge set into short and long edges.
We exhibit a very close relationship between the degeneracy of a graph G and its pandichotomic Euclidean or spherical dimension, that is, the smallest dimension k such that G is pandichotomous in ℝ^k or the sphere 𝒮^k, respectively. First, every d-degenerate graph is pandichotomous in ℝ^d and 𝒮^{d-1} and these bounds are tight for the sphere and for ℝ² and almost tight for ℝ^d, for d ≥ 3. Second, every n-vertex graph that is pandichotomous in ℝ^k has at most μ kn edges, for some absolute constant μ < 7.23. This shows that the pandichotomic Euclidean dimension of any graph is linearly tied to its degeneracy and in the special case k ∈ {1,2} resolves open problems posed by Alam, Kobourov, Pupyrev, and Toeniskoetter.
Further, we characterize which complete bipartite graphs are pandichotomous in ℝ²: These are exactly the K_{m,n} with m ≤ 3 or m = 4 and n ≤ 6. For general bipartite graphs, we can guarantee realizations in ℝ² if the short or the long subgraph is constrained: namely if the short subgraph is outerplanar or a subgraph of a rectangular grid, or if the long subgraph forms a caterpillar.
Cite as
Patrizio Angelini, Sabine Cornelsen, Carolina Haase, Michael Hoffmann, Eleni Katsanou, Fabrizio Montecchiani, Raphael Steiner, and Antonios Symvonis. Geometric Realizations of Dichotomous Ordinal Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{angelini_et_al:LIPIcs.SoCG.2025.9,
author = {Angelini, Patrizio and Cornelsen, Sabine and Haase, Carolina and Hoffmann, Michael and Katsanou, Eleni and Montecchiani, Fabrizio and Steiner, Raphael and Symvonis, Antonios},
title = {{Geometric Realizations of Dichotomous Ordinal Graphs}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {9:1--9:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.9},
URN = {urn:nbn:de:0030-drops-231616},
doi = {10.4230/LIPIcs.SoCG.2025.9},
annote = {Keywords: Ordinal embeddings, geometric graphs, graph representations}
}
Document
Steinhaus Filtration and Stable Paths in the Mapper
Abstract
We define a new filtration called the Steinhaus filtration built from a single cover based on a generalized Steinhaus distance, a generalization of Jaccard distance. The homology persistence module of a Steinhaus filtration with infinitely many cover elements may not be q-tame, even when the covers are in a totally bounded space. While this may pose a challenge to derive stability results, we show that the Steinhaus filtration is stable when the cover is finite. We show that while the Čech and Steinhaus filtrations are not isomorphic in general, they are isomorphic for a finite point set in dimension one. Furthermore, the VR filtration completely determines the 1-skeleton of the Steinhaus filtration in arbitrary dimension.
We then develop a language and theory for stable paths within the Steinhaus filtration. We demonstrate how the framework can be applied to several applications where a standard metric may not be defined but a cover is readily available. We introduce a new perspective for modeling recommendation system datasets. As an example, we look at a movies dataset and we find the stable paths identified in our framework represent a sequence of movies constituting a gentle transition and ordering from one genre to another.
For explainable machine learning, we apply the Mapper algorithm for model induction by building a filtration from a single Mapper complex, and provide explanations in the form of stable paths between subpopulations. For illustration, we build a Mapper complex from a supervised machine learning model trained on the FashionMNIST dataset. Stable paths in the Steinhaus filtration provide improved explanations of relationships between subpopulations of images.
Cite as
Dustin L. Arendt, Matthew Broussard, Bala Krishnamoorthy, Nathaniel Saul, and Amber Thrall. Steinhaus Filtration and Stable Paths in the Mapper. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{arendt_et_al:LIPIcs.SoCG.2025.10,
author = {Arendt, Dustin L. and Broussard, Matthew and Krishnamoorthy, Bala and Saul, Nathaniel and Thrall, Amber},
title = {{Steinhaus Filtration and Stable Paths in the Mapper}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {10:1--10:21},
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.2025.10},
URN = {urn:nbn:de:0030-drops-231625},
doi = {10.4230/LIPIcs.SoCG.2025.10},
annote = {Keywords: Cover and nerve, Jaccard distance, persistence stability, Mapper, recommender systems, explainable machine learning}
}
Document
When Alpha-Complexes Collapse onto Codimension-1 Submanifolds
Abstract
Given a finite set of points P sampling an unknown smooth surface ℳ ⊆ ℝ³, our goal is to triangulate ℳ based solely on P. Assuming ℳ is a smooth orientable submanifold of codimension 1 in ℝ^d, we introduce a simple algorithm, Naive Squash, which simplifies the α-complex of P by repeatedly applying a new type of collapse called vertical relative to ℳ. Naive Squash also has a practical version that does not require knowledge of ℳ. We establish conditions under which both the naive and practical Squash algorithms output a triangulation of ℳ. We provide a bound on the angle formed by triangles in the α-complex with ℳ, yielding sampling conditions on P that are competitive with existing literature for smooth surfaces embedded in ℝ³, while offering a more compartmentalized proof. As a by-product, we obtain that the restricted Delaunay complex of P triangulates ℳ when ℳ is a smooth surface in ℝ³ under weaker conditions than existing ones.
Cite as
Dominique Attali, Mattéo Clémot, Bianca B. Dornelas, and André Lieutier. When Alpha-Complexes Collapse onto Codimension-1 Submanifolds. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{attali_et_al:LIPIcs.SoCG.2025.11,
author = {Attali, Dominique and Cl\'{e}mot, Matt\'{e}o and Dornelas, Bianca B. and Lieutier, Andr\'{e}},
title = {{When Alpha-Complexes Collapse onto Codimension-1 Submanifolds}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {11:1--11:19},
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.2025.11},
URN = {urn:nbn:de:0030-drops-231630},
doi = {10.4230/LIPIcs.SoCG.2025.11},
annote = {Keywords: Submanifold reconstruction, triangulation, abstract simplicial complexes, collapses, convexity}
}
Document
Higher-Order Color Voronoi Diagrams and the Colorful Clarkson-Shor Framework
Abstract
Given a set S of n colored sites, each s ∈ S associated with a distance-to-site function δ_s : ℝ² → ℝ, we consider two distance-to-color functions for each color: one takes the minimum of δ_s for sites s ∈ S in that color and the other takes the maximum. These two sets of distance functions induce two families of higher-order Voronoi diagrams for colors in the plane, namely, the minimal and maximal order-k color Voronoi diagrams, which include various well-studied Voronoi diagrams as special cases. In this paper, we derive an exact upper bound 4k(n-k)-2n on the total number of vertices in both the minimal and maximal order-k color diagrams for a wide class of distance functions δ_s that satisfy certain conditions, including the case of point sites S under convex distance functions and the L_p metric for any 1 ≤ p ≤ ∞. For the L_1 (or, L_∞) metric, and other convex polygonal metrics, we show that the order-k minimal diagram of point sites has O(min{k(n-k), (n-k)²}) complexity, while its maximal counterpart has O(min{k(n-k), k²}) complexity. To obtain these combinatorial results, we extend the Clarkson-Shor framework to colored objects, and demonstrate its application to several fundamental geometric structures, including higher-order color Voronoi diagrams, colored j-facets, and levels in the arrangements of piecewise linear/algebraic curves/surfaces. We also present iterative algorithms to compute higher-order color Voronoi diagrams.
Cite as
Sang Won Bae, Nicolau Oliver, and Evanthia Papadopoulou. Higher-Order Color Voronoi Diagrams and the Colorful Clarkson-Shor Framework. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bae_et_al:LIPIcs.SoCG.2025.12,
author = {Bae, Sang Won and Oliver, Nicolau and Papadopoulou, Evanthia},
title = {{Higher-Order Color Voronoi Diagrams and the Colorful Clarkson-Shor Framework}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {12:1--12:19},
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.2025.12},
URN = {urn:nbn:de:0030-drops-231647},
doi = {10.4230/LIPIcs.SoCG.2025.12},
annote = {Keywords: higher-order Voronoi diagrams, color Voronoi diagrams, Hausdorff Voronoi diagrams, colored j-facets, arrangements, Clarkson-Shor technique}
}
Document
The Erdős-Szekeres Conjecture Revisited
Abstract
The famous and still open Erdős-Szekeres Conjecture from 1935 states that every set of at least 2^{k-2}+1 points in the plane with no three being collinear contains k points in convex position, that is, k points that are vertices of a convex polygon. In this paper, we revisit this conjecture and show several new related results.
First, we prove a relaxed version of the Erdős-Szekeres Conjecture by showing that every set of at least 2^{k-2}+1 points in the plane with no three being collinear contains a split k-gon, a relaxation of k-tuple of points in convex position. Moreover, we show that this is tight, showing that the value 2^{k-2}+1 from the Erdős-Szekeres Conjecture is exactly the right threshold for split k-gons.
We obtain an analogous relaxation in a much more general setting of ordered 3-uniform hypergraphs where we also show that, perhaps surprisingly, a corresponding generalization of the Erdős-Szekeres Conjecture is not true. Finally, we prove the Erdős-Szekeres Conjecture for so-called decomposable sets and provide new constructions of sets of 2^{k-2} points without k points in convex position, generalizing all previously known constructions of such point sets and allowing us to computationally tackle the Erdős-Szekeres Conjecture for large values of k.
Cite as
Jineon Baek and Martin Balko. The Erdős-Szekeres Conjecture Revisited. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{baek_et_al:LIPIcs.SoCG.2025.13,
author = {Baek, Jineon and Balko, Martin},
title = {{The Erd\H{o}s-Szekeres Conjecture Revisited}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {13:1--13:15},
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.2025.13},
URN = {urn:nbn:de:0030-drops-231655},
doi = {10.4230/LIPIcs.SoCG.2025.13},
annote = {Keywords: convex position, Erd\H{o}s-Szekeres theorem, point set}
}
Document
Extremal Betti Numbers and Persistence in Flag Complexes
Abstract
We investigate several problems concerning extremal Betti numbers and persistence in filtrations of flag complexes. For graphs on n vertices, we show that β_k(X(G)) is maximal when G = 𝒯_{n,k+1}, the Turán graph on k+1 partition classes, where X(G) denotes the flag complex of G. Building on this, we construct an edgewise (one edge at a time) filtration 𝒢 = G₁ ⊆ ⋯ ⊆ 𝒯_{n,k+1} for which β_k(X(G_i)) is maximal for all graphs on n vertices and i edges. Moreover, the persistence barcode ℬ_k(X(G)) achieves a maximal number of intervals, and total persistence, among all edgewise filtrations with |E(𝒯_{n,k+1})| edges.
For k = 1, we consider edgewise filtrations of the complete graph K_n. We show that the maximal number of intervals in the persistence barcode is obtained precisely when G_{⌈n/2⌉ ⋅ ⌊n/2⌋} = 𝒯_{n,2}. Among such filtrations, we characterize those achieving maximal total persistence. We further show that no filtration can optimize β₁(X(G_i)) for all i, and conjecture that our filtrations maximize the total persistence over all edgewise filtrations of K_n.
Cite as
Lies Beers and Magnus Bakke Botnan. Extremal Betti Numbers and Persistence in Flag Complexes. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{beers_et_al:LIPIcs.SoCG.2025.14,
author = {Beers, Lies and Bakke Botnan, Magnus},
title = {{Extremal Betti Numbers and Persistence in Flag Complexes}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {14:1--14:18},
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.2025.14},
URN = {urn:nbn:de:0030-drops-231668},
doi = {10.4230/LIPIcs.SoCG.2025.14},
annote = {Keywords: Topological data analysis, Extremal graph theory}
}
Document
When Distances Lie: Euclidean Embeddings in the Presence of Outliers and Distance Violations
Abstract
Distance geometry explores the properties of distance spaces that can be exactly represented as the pairwise Euclidean distances between points in ℝ^d (d ≥ 1), or equivalently, distance spaces that can be isometrically embedded in ℝ^d. In this work, we investigate whether a distance space can be isometrically embedded in ℝ^d after applying a limited number of modifications. Specifically, we focus on two types of modifications: outlier deletion (removing points) and distance modification (adjusting distances between points). The central problem, Euclidean Embedding Editing, asks whether an input distance space on n points can be transformed, using at most k modifications, into a space that is isometrically embeddable in ℝ^d.
We present several fixed-parameter tractable (FPT) and approximation algorithms for this problem. Our first result is an algorithm that solves Euclidean Embedding Editing in time (dk)^𝒪(d+k) + n^𝒪(1). The core subroutine of this algorithm, which is of independent interest, is a polynomial-time method for compressing the input distance space into an equivalent instance of Euclidean Embedding Editing with 𝒪((dk)²) points.
For the special but important case of Euclidean Embedding Editing where only outlier deletions are allowed, we improve the parameter dependence of the FPT algorithm and obtain a running time of min{(d+3)^k, 2^{d+k}} ⋅ n^𝒪(1). Additionally, we provide an FPT-approximation algorithm for this problem, which outputs a set of at most 2 ⋅ Opt outliers in time 2^d ⋅ n^{𝒪(1)}. This 2-approximation algorithm improves upon the previous (3+ε)-approximation algorithm by Sidiropoulos, Wang, and Wang [SODA '17]. Furthermore, we complement our algorithms with hardness results motivating our choice of parameterizations.
Cite as
Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, M. S. Ramanujan, and Saket Saurabh. When Distances Lie: Euclidean Embeddings in the Presence of Outliers and Distance Violations. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bentert_et_al:LIPIcs.SoCG.2025.15,
author = {Bentert, Matthias and Fomin, Fedor V. and Golovach, Petr A. and Ramanujan, M. S. and Saurabh, Saket},
title = {{When Distances Lie: Euclidean Embeddings in the Presence of Outliers and Distance Violations}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {15:1--15:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.15},
URN = {urn:nbn:de:0030-drops-231672},
doi = {10.4230/LIPIcs.SoCG.2025.15},
annote = {Keywords: Parameterized Complexity, Euclidean Embedding, FPT-approximation}
}
Document
Signotopes with Few Plus Signs
Abstract
Arrangements of pseudohyperplanes are widely studied in computational geometry. A rich subclass of pseudohyerplane arrangements, which has gained more attention in recent years, is the so-called signotopes. Introduced by Manin and Schechtman (1989), the higher Bruhat order is a natural order of r-signotopes on n elements, with the signotope corresponding to the cyclic arrangement as the minimal element. In this paper, we show that the lower (and by symmetry upper) levels of this higher Bruhat order contain the same number of elements for a fixed difference n-r. This result implies that given the difference d = n-r and p, the number of one-element extensions of the cyclic arrangement of n hyperplanes in ℝ^d with at most p points on one side of the extending pseudohyperplane does not depend on n, as long as n ≥ d + p.
Cite as
Helena Bergold, Lukas Egeling, and Hung P. Hoang. Signotopes with Few Plus Signs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bergold_et_al:LIPIcs.SoCG.2025.16,
author = {Bergold, Helena and Egeling, Lukas and Hoang, Hung P.},
title = {{Signotopes with Few Plus Signs}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {16:1--16:14},
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.2025.16},
URN = {urn:nbn:de:0030-drops-231681},
doi = {10.4230/LIPIcs.SoCG.2025.16},
annote = {Keywords: flip graph, higher Bruhat order, signotope, counting, Ferrers diagram, one-element extension}
}
Document
Sparse Bounded Hop-Spanners for Geometric Intersection Graphs
Abstract
We present new results on 2- and 3-hop spanners for geometric intersection graphs. These include improved upper and lower bounds for 2- and 3-hop spanners for many geometric intersection graphs in ℝ^d. For example, we show that the intersection graph of n balls in ℝ^d admits a 2-hop spanner of size O^*(n^{3/2 - 1/(2(2⌊d/2⌋ + 1))}) and the intersection graph of n fat axis-parallel boxes in ℝ^d admits a 2-hop spanner of size O(n log^{d+1}n).
Furthermore, we show that the intersection graph of general semi-algebraic objects in ℝ^d admits a 3-hop spanner of size O^*(n^{3/2 - 1/(2(2D-1))}), where D is a parameter associated with the description complexity of the objects. For such families (or more specifically, for tetrahedra in ℝ³), we provide a lower bound of Ω(n^{4/3}). For 3-hop and axis-parallel boxes in ℝ^d, we provide the upper bound O(n log ^{d-1}n) and lower bound Ω(n ({log n}/{log log n})^{d-2}).
Cite as
Sujoy Bhore, Timothy M. Chan, Zhengcheng Huang, Shakhar Smorodinsky, and Csaba D. Tóth. Sparse Bounded Hop-Spanners for Geometric Intersection Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bhore_et_al:LIPIcs.SoCG.2025.17,
author = {Bhore, Sujoy and Chan, Timothy M. and Huang, Zhengcheng and Smorodinsky, Shakhar and T\'{o}th, Csaba D.},
title = {{Sparse Bounded Hop-Spanners for Geometric Intersection Graphs}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {17:1--17:15},
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.2025.17},
URN = {urn:nbn:de:0030-drops-231698},
doi = {10.4230/LIPIcs.SoCG.2025.17},
annote = {Keywords: Geometric Spanners, Geometric Intersection Graphs}
}
Document
Finding a Shortest Curve That Separates Few Objects from Many
Abstract
We present a fixed-parameter tractable (FPT) algorithm to find a shortest curve that encloses a set of k required objects in the plane while paying a penalty for enclosing unwanted objects.
The input is a set of interior-disjoint simple polygons in the plane, where k of the polygons are required to be enclosed and the remaining optional polygons have non-negative penalties. The goal is to find a closed curve that is disjoint from the polygon interiors and encloses the k required polygons, while minimizing the length of the curve plus the penalties of the enclosed optional polygons. If the penalties are high, the output is a shortest curve that separates the required polygons from the others. The problem is NP-hard if k is not fixed, even in very special cases. The runtime of our algorithm is O(3^k n³), where n is the number of vertices of the input polygons.
We extend the result to a graph version of the problem where the input is a connected plane graph with positive edge weights. There are k required faces; the remaining faces are optional and have non-negative penalties. The goal is to find a closed walk in the graph that encloses the k required faces, while minimizing the weight of the walk plus the penalties of the enclosed optional faces. We also consider an inverted version of the problem where the required objects must lie outside the curve. Our algorithms solve some other well-studied problems, such as geometric knapsack.
Cite as
Therese Biedl, Éric Colin de Verdière, Fabrizio Frati, Anna Lubiw, and Günter Rote. Finding a Shortest Curve That Separates Few Objects from Many. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{biedl_et_al:LIPIcs.SoCG.2025.18,
author = {Biedl, Therese and Colin de Verdi\`{e}re, \'{E}ric and Frati, Fabrizio and Lubiw, Anna and Rote, G\"{u}nter},
title = {{Finding a Shortest Curve That Separates Few Objects from Many}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {18:1--18:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.18},
URN = {urn:nbn:de:0030-drops-231701},
doi = {10.4230/LIPIcs.SoCG.2025.18},
annote = {Keywords: Enclosure, curve, separation, weakly simple polygon, Euler tour}
}
Document
Polychromatic Coloring of Tuples in Hypergraphs
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
Polynomial-Time Algorithms for Contiguous Art Gallery and Related Problems
Abstract
We introduce the contiguous art gallery problem which is to guard the boundary of a simple polygon with a minimum number of guards such that each guard covers exactly one contiguous portion of the boundary. Art gallery problems are often NP-hard. In particular, it is NP-hard to minimize the number of guards to see the boundary of a simple polygon, without the contiguity constraint.
This paper is a merge of three concurrent works [Ahmad Biniaz et al., 2024; Magnus Christian Ring Merrild et al., 2024; Eliot W. Robson et al., 2024] each showing that (surprisingly) the contiguous art gallery problem is solvable in polynomial time. The common idea of all three approaches is developing a greedy function that maps a point on the boundary to the furthest point on the boundary so that the contiguous interval along the boundary between them could be guarded by one guard. Repeatedly applying this function immediately leads to an OPT+1 approximation. By studying this greedy algorithm, we present three different approaches that achieve an optimal solution. The first and second approach apply this greedy algorithm from different points on the boundary that could be found in advance or on the fly while traversing along the boundary (respectively). The third approach represents this function as a piecewise linear rational function, which can be reduced to an abstract arc cover problem involving infinite families of arcs. We identify other problems that can be represented by similar functions, and solve them via the third approach.
From the combinatorial point of view, we show that any n-vertex polygon can be guarded by at most ⌊(n-2)/2⌋ guards. This bound is tight because there are polygons that require this many guards.
Cite as
Ahmad Biniaz, Anil Maheshwari, Magnus Christian Ring Merrild, Joseph S. B. Mitchell, Saeed Odak, Valentin Polishchuk, Eliot W. Robson, Casper Moldrup Rysgaard, Jens Kristian Refsgaard Schou, Thomas Shermer, Jack Spalding-Jamieson, Rolf Svenning, and Da Wei Zheng. Polynomial-Time Algorithms for Contiguous Art Gallery and Related Problems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 20:1-20:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{biniaz_et_al:LIPIcs.SoCG.2025.20,
author = {Biniaz, Ahmad and Maheshwari, Anil and Merrild, Magnus Christian Ring and Mitchell, Joseph S. B. and Odak, Saeed and Polishchuk, Valentin and Robson, Eliot W. and Rysgaard, Casper Moldrup and Schou, Jens Kristian Refsgaard and Shermer, Thomas and Spalding-Jamieson, Jack and Svenning, Rolf and Zheng, Da Wei},
title = {{Polynomial-Time Algorithms for Contiguous Art Gallery and Related Problems}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {20:1--20:21},
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.2025.20},
URN = {urn:nbn:de:0030-drops-231720},
doi = {10.4230/LIPIcs.SoCG.2025.20},
annote = {Keywords: Art Gallery Problem, Computational Geometry, Combinatorics, Discrete Algorithms}
}
Document
Structure and Independence in Hyperbolic Uniform Disk Graphs
Abstract
We consider intersection graphs of disks of radius r in the hyperbolic plane. Unlike the Euclidean setting, these graph classes are different for different values of r, where very small r corresponds to an almost-Euclidean setting and r ∈ Ω(log n) corresponds to a firmly hyperbolic setting. We observe that larger values of r create simpler graph classes, at least in terms of separators and the computational complexity of the Independent Set problem.
First, we show that intersection graphs of disks of radius r in the hyperbolic plane can be separated with 𝒪((1+1/r)log n) cliques in a balanced manner. Our second structural insight concerns Delaunay complexes in the hyperbolic plane and may be of independent interest. We show that for any set S of n points with pairwise distance at least 2r in the hyperbolic plane, the corresponding Delaunay complex has outerplanarity 1+𝒪((log n)/r), which implies a similar bound on the balanced separators and treewidth of such Delaunay complexes.
Using this outerplanarity (and treewidth) bound we prove that Independent Set can be solved in n^𝒪(1+(log n)/r) time. The algorithm is based on dynamic programming on some unknown sphere cut decomposition that is based on the solution. The resulting algorithm is a far-reaching generalization of a result of Kisfaludi-Bak (SODA 2020), and it is tight under the Exponential Time Hypothesis. In particular, Independent Set is polynomial-time solvable in the firmly hyperbolic setting of r ∈ Ω(log n). Finally, in the case when the disks have ply (depth) at most 𝓁, we give a PTAS for Maximum Independent Set that has only quasi-polynomial dependence on 1/ε and 𝓁. Our PTAS is a further generalization of our exact algorithm.
Cite as
Thomas Bläsius, Jean-Pierre von der Heydt, Sándor Kisfaludi-Bak, Marcus Wilhelm, and Geert van Wordragen. Structure and Independence in Hyperbolic Uniform Disk Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{blasius_et_al:LIPIcs.SoCG.2025.21,
author = {Bl\"{a}sius, Thomas and von der Heydt, Jean-Pierre and Kisfaludi-Bak, S\'{a}ndor and Wilhelm, Marcus and van Wordragen, Geert},
title = {{Structure and Independence in Hyperbolic Uniform Disk Graphs}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {21:1--21:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.21},
URN = {urn:nbn:de:0030-drops-231731},
doi = {10.4230/LIPIcs.SoCG.2025.21},
annote = {Keywords: hyperbolic geometry, unit disk graphs, independent set, treewidth}
}
Document
Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D
Abstract
The Fréchet distance is a computational mainstay for comparing polygonal curves. The Fréchet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in space. It is defined as the minimum Fréchet distance that arises from allowing arbitrary translations of the input curves. This problem and numerous variants of the Fréchet distance under some transformations have been studied, with more work concentrating on the discrete Fréchet distance, leaving a significant gap between the discrete and continuous versions of the Fréchet distance under transformations. Our contribution is twofold: First, we present an algorithm for the Fréchet distance under translation on 1-dimensional curves of complexity n with a running time of 𝒪(n^{8/3} log³ n). To achieve this, we develop a novel framework for the problem for 1-dimensional curves, which also applies to other scenarios and leads to our second contribution. We present an algorithm with the same running time of 𝒪(n^{8/3} log³ n) for the Fréchet distance under scaling for 1-dimensional curves. For both algorithms we match the running times of the discrete case and improve the previously best known bounds of 𝒪̃(n⁴). Our algorithms rely on technical insights but are conceptually simple, essentially reducing the continuous problem to the discrete case across different length scales.
Cite as
Lotte Blank, Jacobus Conradi, Anne Driemel, Benedikt Kolbe, André Nusser, and Marena Richter. Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{blank_et_al:LIPIcs.SoCG.2025.22,
author = {Blank, Lotte and Conradi, Jacobus and Driemel, Anne and Kolbe, Benedikt and Nusser, Andr\'{e} and Richter, Marena},
title = {{Transforming Dogs on the Line: On the Fr\'{e}chet Distance Under Translation or Scaling in 1D}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {22:1--22:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.22},
URN = {urn:nbn:de:0030-drops-231746},
doi = {10.4230/LIPIcs.SoCG.2025.22},
annote = {Keywords: Fr\'{e}chet distance under translation, Fr\'{e}chet distance under scaling, time series, shape matching}
}
Document
On the Twin-Width of Smooth Manifolds
Abstract
Building on Whitney’s classical method of triangulating smooth manifolds, we show that every compact d-dimensional smooth manifold admits a triangulation with dual graph of twin-width at most d^O(d). In particular, it follows that every compact 3-manifold has a triangulation with dual graph of bounded twin-width. This is in sharp contrast to the case of treewidth, where for any natural number n there exists a closed 3-manifold such that every triangulation thereof has dual graph with treewidth at least n. To establish this result, we bound the twin-width of the dual graph of the d-skeleton of the second barycentric subdivision of the 2d-dimensional hypercubic honeycomb. We also show that every compact, piecewise-linear (hence smooth) d-dimensional manifold has triangulations where the dual graph has an arbitrarily large twin-width.
Cite as
Édouard Bonnet and Kristóf Huszár. On the Twin-Width of Smooth Manifolds. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bonnet_et_al:LIPIcs.SoCG.2025.23,
author = {Bonnet, \'{E}douard and Husz\'{a}r, Krist\'{o}f},
title = {{On the Twin-Width of Smooth Manifolds}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {23:1--23:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.23},
URN = {urn:nbn:de:0030-drops-231752},
doi = {10.4230/LIPIcs.SoCG.2025.23},
annote = {Keywords: Smooth manifolds, triangulations, twin-width, Whitney embedding theorem, structural graph parameters, computational topology}
}
Document
An 11/6-Approximation Algorithm for Vertex Cover on String Graphs
Abstract
We present a 1.8334-approximation algorithm for Vertex Cover on string graphs given with a representation, which takes polynomial time in the size of the representation; the exact approximation factor is 11/6. Recently, the barrier of 2 was broken by Lokshtanov, Panolan, Saurabh, Xue, and Zehavi [SoGC '24] with a 1.9999-approximation algorithm. Thus we increase by three orders of magnitude the distance of the approximation ratio to the trivial bound of 2. Our algorithm is very simple. The intricacies reside in its analysis, where we mainly establish that string graphs without odd cycles of length at most 11 are 8-colorable. Previously, Chudnovsky, Scott, and Seymour [JCTB '21] showed that string graphs without odd cycles of length at most 7 are 80-colorable, and string graphs without odd cycles of length at most 5 have bounded chromatic number.
Cite as
Édouard Bonnet and Paweł Rzążewski. An 11/6-Approximation Algorithm for Vertex Cover on String Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bonnet_et_al:LIPIcs.SoCG.2025.24,
author = {Bonnet, \'{E}douard and Rz\k{a}\.{z}ewski, Pawe{\l}},
title = {{An 11/6-Approximation Algorithm for Vertex Cover on String Graphs}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {24:1--24:15},
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.2025.24},
URN = {urn:nbn:de:0030-drops-231764},
doi = {10.4230/LIPIcs.SoCG.2025.24},
annote = {Keywords: Approximation algorithms, string graphs, Vertex Cover, Coloring, odd girth}
}
Document
Approximating Klee’s Measure Problem and a Lower Bound for Union Volume Estimation
Abstract
Union volume estimation is a classical algorithmic problem. Given a family of objects O₁,…,O_n ⊂ ℝ^d, we want to approximate the volume of their union. In the special case where all objects are boxes (also called hyperrectangles) this is known as Klee’s measure problem. The state-of-the-art (1+ε)-approximation algorithm [Karp, Luby, Madras '89] for union volume estimation as well as Klee’s measure problem in constant dimension d uses a total of O(n/ε²) queries of three types: (i) determine the volume of O_i; (ii) sample a point uniformly at random from O_i; and (iii) ask whether a given point is contained in O_i.
First, we show that if an algorithm learns about the objects only through these types of queries, then Ω(n/ε²) queries are necessary. In this sense, the complexity of [Karp, Luby, Madras '89] is optimal. Our lower bound holds even if the objects are equiponderous axis-aligned polygons in ℝ², if the containment query allows arbitrary (not necessarily sampled) points, and if the algorithm can spend arbitrary time and space examining the query responses.
Second, we provide a more efficient approximation algorithm for Klee’s measure problem, which improves the running time from O(n/ε²) to O((n+1/ε²) ⋅ log^{O(d)} (n)). We circumvent our lower bound by exploiting the geometry of boxes in various ways: (1) We sort the boxes into classes of similar shapes after inspecting their corner coordinates. (2) With orthogonal range searching, we show how to sample points from the union of boxes in each class, and how to merge samples from different classes. (3) We bound the amount of wasted work by arguing that most pairs of classes have a small intersection.
Cite as
Karl Bringmann, Kasper Green Larsen, André Nusser, Eva Rotenberg, and Yanheng Wang. Approximating Klee’s Measure Problem and a Lower Bound for Union Volume Estimation. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2025.25,
author = {Bringmann, Karl and Larsen, Kasper Green and Nusser, Andr\'{e} and Rotenberg, Eva and Wang, Yanheng},
title = {{Approximating Klee’s Measure Problem and a Lower Bound for Union Volume Estimation}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {25:1--25:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.25},
URN = {urn:nbn:de:0030-drops-231778},
doi = {10.4230/LIPIcs.SoCG.2025.25},
annote = {Keywords: approximation, volume of union, union of objects, query complexity}
}
Document
Geometric Spanners of Bounded Tree-Width
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}
}
Document
Computing Oriented Spanners and Their Dilation
Abstract
Given a point set P in a metric space and a real number t ≥ 1, an oriented t-spanner is an oriented graph G = (P, E), where for every pair of distinct points p and q in P, the shortest oriented closed walk in G that contains p and q is at most a factor t longer than the perimeter of the smallest triangle in P containing p and q. The oriented dilation of a graph G is the minimum t for which G is an oriented t-spanner.
For arbitrary point sets of size n in ℝ^d, where d ≥ 2 is a constant, the only known oriented spanner construction is an oriented 2-spanner with binom(n,2) edges. Moreover, there exists a set P of four points in the plane, for which the oriented dilation is larger than 1.46, for any oriented graph on P.
We present the first algorithm that computes, in Euclidean space, a sparse oriented spanner whose oriented dilation is bounded by a constant. More specifically, for any set of n points in ℝ^d, where d is a constant, we construct an oriented (2+ε)-spanner with 𝒪(n) edges in 𝒪(n log n) time and 𝒪(n) space. Our construction uses the well-separated pair decomposition and an algorithm that computes a (1+ε)-approximation of the minimum-perimeter triangle in P containing two given query points in 𝒪(log n) time.
While our algorithm is based on first computing a suitable undirected graph and then orienting it, we show that, in general, computing the orientation of an undirected graph that minimises its oriented dilation is NP-hard, even for point sets in the Euclidean plane.
We further prove that even if the oriented graph is already given, computing its oriented dilation is APSP-hard for points in a general metric space. We complement this result with an algorithm that approximates the oriented dilation of a given graph in subcubic time for point sets in ℝ^d, where d is a constant.
Cite as
Kevin Buchin, Antonia Kalb, Anil Maheshwari, Saeed Odak, Carolin Rehs, Michiel Smid, and Sampson Wong. Computing Oriented Spanners and Their Dilation. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{buchin_et_al:LIPIcs.SoCG.2025.27,
author = {Buchin, Kevin and Kalb, Antonia and Maheshwari, Anil and Odak, Saeed and Rehs, Carolin and Smid, Michiel and Wong, Sampson},
title = {{Computing Oriented Spanners and Their Dilation}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {27:1--27:17},
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.2025.27},
URN = {urn:nbn:de:0030-drops-231792},
doi = {10.4230/LIPIcs.SoCG.2025.27},
annote = {Keywords: spanner, oriented graph, dilation, orientation, well-separated pair decomposition, minimum-perimeter triangle}
}
Document
Small Triangulations of 4-Manifolds and the 4-Manifold Census
Abstract
We present a framework to classify PL-types of large censuses of triangulated 4-manifolds, which we use to classify the PL-types of all triangulated 4-manifolds with up to 6 pentachora. This is successful except for triangulations homeomorphic to the 4-sphere, CP², and the rational homology sphere QS⁴(2), where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations - which we deem interesting in their own rights.
Cite as
Rhuaidi Antonio Burke, Benjamin A. Burton, and Jonathan Spreer. Small Triangulations of 4-Manifolds and the 4-Manifold Census. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{burke_et_al:LIPIcs.SoCG.2025.28,
author = {Burke, Rhuaidi Antonio and Burton, Benjamin A. and Spreer, Jonathan},
title = {{Small Triangulations of 4-Manifolds and the 4-Manifold Census}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {28:1--28:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.28},
URN = {urn:nbn:de:0030-drops-231805},
doi = {10.4230/LIPIcs.SoCG.2025.28},
annote = {Keywords: computational low-dimensional topology, triangulations, census of triangulations, 4-manifolds, PL standard 4-sphere, Pachner graph, mathematical software, experiments in low-dimensional topology}
}
Document
Sparsification of the Generalized Persistence Diagrams for Scalability Through Gradient Descent
Abstract
The generalized persistence diagram (GPD) is a natural extension of the classical persistence barcode to the setting of multi-parameter persistence and beyond. The GPD is defined as an integer-valued function whose domain is the set of intervals in the indexing poset of a persistence module, and is known to be able to capture richer topological information than its single-parameter counterpart. However, computing the GPD is computationally prohibitive due to the sheer size of the interval set. Restricting the GPD to a subset of intervals provides a way to manage this complexity, compromising discriminating power to some extent. However, identifying and computing an effective restriction of the domain that minimizes the loss of discriminating power remains an open challenge.
In this work, we introduce a novel method for optimizing the domain of the GPD through gradient descent optimization. To achieve this, we introduce a loss function tailored to optimize the selection of intervals, balancing computational efficiency and discriminative accuracy. The design of the loss function is based on the known erosion stability property of the GPD. We showcase the efficiency of our sparsification method for dataset classification in supervised machine learning. Experimental results demonstrate that our sparsification method significantly reduces the time required for computing the GPDs associated to several datasets, while maintaining classification accuracies comparable to those achieved using full GPDs. Our method thus opens the way for the use of GPD-based methods to applications at an unprecedented scale.
Cite as
Mathieu Carrière, Seunghyun Kim, and Woojin Kim. Sparsification of the Generalized Persistence Diagrams for Scalability Through Gradient Descent. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{carriere_et_al:LIPIcs.SoCG.2025.29,
author = {Carri\`{e}re, Mathieu and Kim, Seunghyun and Kim, Woojin},
title = {{Sparsification of the Generalized Persistence Diagrams for Scalability Through Gradient Descent}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {29:1--29:17},
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.2025.29},
URN = {urn:nbn:de:0030-drops-231810},
doi = {10.4230/LIPIcs.SoCG.2025.29},
annote = {Keywords: Multi-parameter persistent homology, Generalized persistence diagram, Generalized rank invariant, Non-convex optimization, Gradient descent}
}
Document
Computation of Toroidal Schnyder Woods Made Simple and Fast: From Theory to Practice
Abstract
We consider the problem of computing Schnyder woods for graphs embedded on the torus. We design simple linear-time algorithms based on canonical orderings that compute toroidal Schnyder woods for simple toroidal triangulations. The Schnyder woods computed by one of our algorithm are crossing and satisfy an additional structural property: at least two of the mono-chromatic components of the Schnyder wood are connected. We also exhibit experimental results empirically confirming three conjectures involving the structure of toroidal and higher genus Schnyder woods.
Cite as
Luca Castelli Aleardi, Eric Fusy, Jyh-Chwen Ko, and Razvan-Stefan Puscasu. Computation of Toroidal Schnyder Woods Made Simple and Fast: From Theory to Practice. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{castellialeardi_et_al:LIPIcs.SoCG.2025.30,
author = {Castelli Aleardi, Luca and Fusy, Eric and Ko, Jyh-Chwen and Puscasu, Razvan-Stefan},
title = {{Computation of Toroidal Schnyder Woods Made Simple and Fast: From Theory to Practice}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {30:1--30:19},
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.2025.30},
URN = {urn:nbn:de:0030-drops-231825},
doi = {10.4230/LIPIcs.SoCG.2025.30},
annote = {Keywords: Schnyder woods, toroidal triangulations, canonical ordering}
}
Document
A Linear Time Algorithm for the Maximum Overlap of Two Convex Polygons Under Translation
Abstract
Given two convex polygons P and Q with n and m edges, the maximum overlap problem is to find a translation of P that maximizes the area of its intersection with Q. We give the first randomized algorithm for this problem with linear running time. Our result improves the previous two-and-a-half-decades-old algorithm by de Berg, Cheong, Devillers, van Kreveld, and Teillaud (1998), which ran in O((n+m)log(n+m)) time, as well as multiple recent algorithms given for special cases of the problem.
Cite as
Timothy M. Chan and Isaac M. Hair. A Linear Time Algorithm for the Maximum Overlap of Two Convex Polygons Under Translation. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chan_et_al:LIPIcs.SoCG.2025.31,
author = {Chan, Timothy M. and Hair, Isaac M.},
title = {{A Linear Time Algorithm for the Maximum Overlap of Two Convex Polygons Under Translation}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {31:1--31:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.31},
URN = {urn:nbn:de:0030-drops-231832},
doi = {10.4230/LIPIcs.SoCG.2025.31},
annote = {Keywords: Convex polygons, shape matching, prune-and-search, parametric search}
}
Document
Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-And-Bifurcate Technique
Abstract
In a series of papers, Avraham, Filtser, Kaplan, Katz, and Sharir (SoCG'14), Kaplan, Katz, Saban, and Sharir (ESA'23), and Katz, Saban, and Sharir (ESA'24) studied a class of geometric optimization problems - including reverse shortest path in unweighted and weighted unit-disk graphs, discrete Fréchet distance with one-sided shortcuts, and reverse shortest path in visibility graphs on 1.5-dimensional terrains - for which standard parametric search does not work well due to a lack of efficient parallel algorithms for the corresponding decision problems. The best currently known algorithms for all the above problems run in O^*(n^{6/5}) = O^*(n^{1.2}) time (ignoring subpolynomial factors), and they were obtained using a technique called shrink-and-bifurcate. We improve the running time to Õ(n^{8/7}) ≈ O(n^{1.143}) for these problems. Furthermore, specifically for reverse shortest path in unweighted unit-disk graphs, we improve the running time further to Õ(n^{9/8}) = Õ(n^{1.125}).
Cite as
Timothy M. Chan and Zhengcheng Huang. Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-And-Bifurcate Technique. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chan_et_al:LIPIcs.SoCG.2025.32,
author = {Chan, Timothy M. and Huang, Zhengcheng},
title = {{Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-And-Bifurcate Technique}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {32:1--32:14},
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.2025.32},
URN = {urn:nbn:de:0030-drops-231845},
doi = {10.4230/LIPIcs.SoCG.2025.32},
annote = {Keywords: Geometric optimization problems, parametric search, shortest path, disk graphs, Fr\'{e}chet distance, visibility, distance selection, randomized algorithms}
}
Document
On Zarankiewicz’s Problem for Intersection Hypergraphs of Geometric Objects
Abstract
In this paper we study the hypergraph Zarankiewicz’s problem in a geometric setting - for r-partite intersection hypergraphs of families of geometric objects. Our main results are essentially sharp bounds for families of axis-parallel boxes in ℝ^d and families of pseudo-discs. For axis-parallel boxes, we obtain the sharp bound O_{d,t}(n^{r-1}((log n)/(log log n))^{d-1}). The best previous bound was larger by a factor of about (log n)^{d(2^{r-1}-2)}. For pseudo-discs, we obtain the bound O_t(n^{r-1}(log n)^{r-2}), which is sharp up to logarithmic factors. As this hypergraph has no algebraic structure, no improvement of Erdős' 60-year-old O(n^{r-(1/t^{r-1})}) bound was known for this setting. Futhermore, even in the special case of discs for which the semialgebraic structure can be used, our result improves the best known result by a factor of Ω̃(n^{(2r-2)/(3r-2)}).
To obtain our results, we use the recently improved results for the graph Zarankiewicz’s problem in the corresponding settings, along with a variety of combinatorial and geometric techniques, including shallow cuttings, biclique covers, transversals, and planarity.
Cite as
Timothy M. Chan, Chaya Keller, and Shakhar Smorodinsky. On Zarankiewicz’s Problem for Intersection Hypergraphs of Geometric Objects. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chan_et_al:LIPIcs.SoCG.2025.33,
author = {Chan, Timothy M. and Keller, Chaya and Smorodinsky, Shakhar},
title = {{On Zarankiewicz’s Problem for Intersection Hypergraphs of Geometric Objects}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {33:1--33:14},
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.2025.33},
URN = {urn:nbn:de:0030-drops-231850},
doi = {10.4230/LIPIcs.SoCG.2025.33},
annote = {Keywords: Zarankiewicz’s Problem, hypergraphs, intersection graphs, axis-parallel boxes, pseudo-discs}
}
Document
Simplification of Trajectory Streams
Abstract
While there are software systems that simplify trajectory streams on the fly, few curve simplification algorithms with quality guarantees fit the streaming requirements. We present streaming algorithms for two such problems under the Fréchet distance d_F in ℝ^d for some constant d ≥ 2.
Consider a polygonal curve τ in ℝ^d in a stream. We present a streaming algorithm that, for any ε ∈ (0,1) and δ > 0, produces a curve σ such that d_F(σ,τ[v₁,v_i]) ≤ (1+ε)δ and |σ| ≤ 2 opt-2, where τ[v₁,v_i] is the prefix in the stream so far, and opt = min{|σ'|: d_F(σ',τ[v₁,v_i]) ≤ δ}. Let α = 2(d-1)⌊d/2⌋² + d. The working storage is O(ε^{-α}). Each vertex is processed in O(ε^{-α} log 1/ε) time for d ∈ {2,3} and O(ε^{-α}) time for d ≥ 4 . Thus, the whole τ can be simplified in O(ε^{-α}|τ| log 1/ε) time. Ignoring polynomial factors in 1/ε, this running time is a factor |τ| faster than the best static algorithm that offers the same guarantees.
We present another streaming algorithm that, for any integer k ≥ 2 and any ε ∈ (0,1/17), maintains a curve σ such that |σ| ≤ 2k-2 and d_F(σ,τ[v₁,v_i]) ≤ (1+ε) ⋅ min{d_F(σ',τ[v₁,v_i]): |σ'| ≤ k}, where τ[v₁,v_i] is the prefix in the stream so far. The working storage is O((kε^{-1}+ε^{-(α+1)})log 1/(ε)). Each vertex is processed in O(kε^{-(α+1)}log²1/(ε)) time for d ∈ {2,3} and O(kε^{-(α+1)} log 1/ε) time for d ≥ 4.
Cite as
Siu-Wing Cheng, Haoqiang Huang, and Le Jiang. Simplification of Trajectory Streams. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{cheng_et_al:LIPIcs.SoCG.2025.34,
author = {Cheng, Siu-Wing and Huang, Haoqiang and Jiang, Le},
title = {{Simplification of Trajectory Streams}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {34:1--34:14},
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.2025.34},
URN = {urn:nbn:de:0030-drops-231860},
doi = {10.4230/LIPIcs.SoCG.2025.34},
annote = {Keywords: streaming algorithm, curve simplification, Fr\'{e}chet distance}
}
Document
A Theory of Sub-Barcodes
Abstract
The primary tool in topological data analysis (TDA) is persistent homology, which involves computing a barcode - often from point-cloud or scalar field data - that serves as a topological signature for the underlying function. In this work, we introduce sub-barcodes and show how they arise naturally from factorizations of persistence module homomorphisms. We show that, as a partial order induced by factorizations, the relation of being a sub-barcode is strictly stronger than the rank invariant, and we apply sub-barcode theory to the problem of inferring information about the barcode of an unknown Lipschitz function from samples. The advantage of this approach is that it permits strong guarantees - with no noise - while requiring no sampling assumptions, and the resulting barcode is guaranteed to be a sub-barcode of every Lipschitz function that agrees with the data. We also present an algorithmic theory that allows for the efficient approximation of sub-barcodes using filtered Delaunay triangulations for Euclidean inputs.
Cite as
Oliver A. Chubet, Kirk P. Gardner, and Donald R. Sheehy. A Theory of Sub-Barcodes. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chubet_et_al:LIPIcs.SoCG.2025.35,
author = {Chubet, Oliver A. and Gardner, Kirk P. and Sheehy, Donald R.},
title = {{A Theory of Sub-Barcodes}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {35:1--35:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.35},
URN = {urn:nbn:de:0030-drops-231873},
doi = {10.4230/LIPIcs.SoCG.2025.35},
annote = {Keywords: Topology, Topological Data Analysis, Persistent Homology, Persistence Modules, Barcodes, Sub-barcodes, Factorizations, Lipschitz Extensions}
}
Document
Strongly Sublinear Separators and Bounded Asymptotic Dimension for Sphere Intersection Graphs
Abstract
In this paper, we consider the class 𝒞^d of sphere intersection graphs in R^d for d ≥ 2. We show that for each integer t, the class of all graphs in 𝒞^d that exclude K_{t,t} as a subgraph has strongly sublinear separators. We also prove that 𝒞^d has asymptotic dimension at most 2d+2.
Cite as
James Davies, Agelos Georgakopoulos, Meike Hatzel, and Rose McCarty. Strongly Sublinear Separators and Bounded Asymptotic Dimension for Sphere Intersection Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{davies_et_al:LIPIcs.SoCG.2025.36,
author = {Davies, James and Georgakopoulos, Agelos and Hatzel, Meike and McCarty, Rose},
title = {{Strongly Sublinear Separators and Bounded Asymptotic Dimension for Sphere Intersection Graphs}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {36:1--36:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.36},
URN = {urn:nbn:de:0030-drops-231881},
doi = {10.4230/LIPIcs.SoCG.2025.36},
annote = {Keywords: Intersection graphs, strongly sublinear separators, asymptotic dimension}
}
Document
Nearest Neighbor Searching in a Dynamic Simple Polygon
Abstract
In the nearest neighbor problem, we are given a set S of point sites that we want to store such that we can find the nearest neighbor of a (new) query point efficiently. In the dynamic version of the problem, the goal is to design a data structure that supports both efficient queries and updates, i.e. insertions and deletions in S. This problem has been widely studied in various settings, ranging from points in the plane to more general distance measures and even points within simple polygons. When the sites do not live in the plane but in some domain, another dynamic problem arises: what happens if not the sites, but the domain itself is subject to updates?
Updating sites often results in local changes to the solution or data structure, while updating the domain may incur many global changes. For example, in the closest pair problem, inserting a point only requires us to check if this point is in the new closest pair, while updating the domain might change the distances between most pairs of points in our set. Presumably, this is the reason that this form of dynamization has received much less attention. Only some basic problems, such as shortest paths and ray shooting, have been studied in this setting.
Here, we tackle the nearest neighbor problem in a dynamic simple polygon. We allow insertions into both the set of sites and the polygon. An insertion in the polygon is the addition of a line segment starting at the boundary of the polygon. We present a near-linear size -in both the number of sites and the complexity of the polygon- data structure with sublinear update and query time. This is the first nearest neighbor data structure that allows for updates to the domain.
Cite as
Sarita de Berg and Frank Staals. Nearest Neighbor Searching in a Dynamic Simple Polygon. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{deberg_et_al:LIPIcs.SoCG.2025.37,
author = {de Berg, Sarita and Staals, Frank},
title = {{Nearest Neighbor Searching in a Dynamic Simple Polygon}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {37:1--37:15},
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.2025.37},
URN = {urn:nbn:de:0030-drops-231896},
doi = {10.4230/LIPIcs.SoCG.2025.37},
annote = {Keywords: dynamic data structure, simple polygon, geodesic distance, nearest neighbor}
}
Document
An Algorithm for Tambara-Yamagami Quantum Invariants of 3-Manifolds, Parameterized by the First Betti Number
Abstract
Quantum topology provides various frameworks for defining and computing invariants of manifolds inspired by quantum theory. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we analyze the computational complexity of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. While these categories are the simplest source of state sum invariants beyond finite abelian groups (whose invariants can be computed in polynomial time) their computational complexities are yet to be fully understood. We first establish that the invariants arising from even the smallest Tambara-Yamagami categories are #P-hard to compute, so that one expects the same to be true of the whole family. Our main result is then the existence of a fixed parameter tractable algorithm to compute these 3-manifold invariants, where the parameter is the first Betti number of the 3-manifold with ℤ/2ℤ coefficients.
Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is computable in polynomial time. Thus while one generally expects quantum invariants to be hard to compute classically, our results suggest that the hardness of computing state sum invariants from Tambara-Yamagami categories arises from classical 3-manifold topology rather than the quantum nature of the algebraic input.
Cite as
Colleen Delaney, Clément Maria, and Eric Samperton. An Algorithm for Tambara-Yamagami Quantum Invariants of 3-Manifolds, Parameterized by the First Betti Number. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{delaney_et_al:LIPIcs.SoCG.2025.38,
author = {Delaney, Colleen and Maria, Cl\'{e}ment and Samperton, Eric},
title = {{An Algorithm for Tambara-Yamagami Quantum Invariants of 3-Manifolds, Parameterized by the First Betti Number}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {38:1--38:15},
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.2025.38},
URN = {urn:nbn:de:0030-drops-231901},
doi = {10.4230/LIPIcs.SoCG.2025.38},
annote = {Keywords: 3-manifold, quantum invariant, fixed parameter tractable algorithm, topological parameter, Gauss sums, topological quantum field theory}
}
Document
Tiling with Three Polygons Is Undecidable
Abstract
We prove that the following problem is co-RE-complete and thus undecidable: given three simple polygons, is there a tiling of the plane where every tile is an isometry of one of the three polygons (either allowing or forbidding reflections)? This result improves on the best previous construction which requires five polygons.
Cite as
Erik D. Demaine and Stefan Langerman. Tiling with Three Polygons Is Undecidable. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{demaine_et_al:LIPIcs.SoCG.2025.39,
author = {Demaine, Erik D. and Langerman, Stefan},
title = {{Tiling with Three Polygons Is Undecidable}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {39:1--39:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.39},
URN = {urn:nbn:de:0030-drops-231913},
doi = {10.4230/LIPIcs.SoCG.2025.39},
annote = {Keywords: plane tilings, polygons, undecidability, co-RE}
}
Document
Apex Representatives
Abstract
Given a zigzag filtration, we want to find its barcode representatives, i.e., a compatible choice of bases for the homology groups that diagonalize the linear maps in the zigzag. To achieve this, we convert the input zigzag to a levelset zigzag of a real-valued function. This function generates a Mayer-Vietoris pyramid of spaces, which generates an infinite strip of homology groups. We call the origins of indecomposable (diamond) summands of this strip their apexes and give an algorithm to find representative cycles in these apexes from ordinary persistence computation. The resulting representatives map back to the levelset zigzag and thus yield barcode representatives for the input zigzag. Our algorithm for lifting a p-dimensional cycle from ordinary persistence to an apex representative takes O(p ⋅ m log m) time. From this we can recover zigzag representatives in time O(log m + C), where C is the size of the output.
Cite as
Tamal K. Dey, Tao Hou, and Dmitriy Morozov. Apex Representatives. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{dey_et_al:LIPIcs.SoCG.2025.40,
author = {Dey, Tamal K. and Hou, Tao and Morozov, Dmitriy},
title = {{Apex Representatives}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {40:1--40:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.40},
URN = {urn:nbn:de:0030-drops-231927},
doi = {10.4230/LIPIcs.SoCG.2025.40},
annote = {Keywords: zigzag persistent homology, Mayer-Vietoris pyramid, cycle representatives}
}
Document
Decomposing Multiparameter Persistence Modules
Abstract
Dey and Xin (J.Appl.Comput.Top., 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed-parameter tractable with respect to the maximal number of relations of the same degree and with further optimisation we obtain an O(n³) time algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida, the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.
Cite as
Tamal K. Dey, Jan Jendrysiak, and Michael Kerber. Decomposing Multiparameter Persistence Modules. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{dey_et_al:LIPIcs.SoCG.2025.41,
author = {Dey, Tamal K. and Jendrysiak, Jan and Kerber, Michael},
title = {{Decomposing Multiparameter Persistence Modules}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {41:1--41:19},
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.2025.41},
URN = {urn:nbn:de:0030-drops-231939},
doi = {10.4230/LIPIcs.SoCG.2025.41},
annote = {Keywords: Topological Data Analysis, Multiparameter Persistence Modules, Persistence, Decomposition}
}
Document
Range Counting Oracles for Geometric Problems
Abstract
In this paper, we study estimators for geometric optimization problems in the sublinear geometric model. In this model, we have oracle access to a point set with size n in a discrete space [Δ]^d, where queries can be made to an oracle that responds to orthogonal range counting requests. The query complexity of an optimization problem is measured by the number of oracle queries required to compute an estimator for the problem. We investigate two problems in this framework, the Euclidean Minimum Spanning Tree (MST) and Earth Mover Distance (EMD). For EMD, we show the existence of an estimator that approximates the cost of EMD with O(log Δ)-relative error and O(nΔ/(s^{1+1/d}))-additive error using O(s polylog Δ) range counting queries for any parameter s with 1 ≤ s ≤ n. Moreover, we prove that this bound is tight. For MST, we demonstrate that the weight of MST can be estimated within a factor of (1 ± ε) using Õ(√n) range counting queries.
Cite as
Anne Driemel, Morteza Monemizadeh, Eunjin Oh, Frank Staals, and David P. Woodruff. Range Counting Oracles for Geometric Problems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 42:1-42:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{driemel_et_al:LIPIcs.SoCG.2025.42,
author = {Driemel, Anne and Monemizadeh, Morteza and Oh, Eunjin and Staals, Frank and Woodruff, David P.},
title = {{Range Counting Oracles for Geometric Problems}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {42:1--42:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.42},
URN = {urn:nbn:de:0030-drops-231941},
doi = {10.4230/LIPIcs.SoCG.2025.42},
annote = {Keywords: Range counting oracles, minimum spanning trees, Earth Mover’s Distance}
}
Document
On Spheres with k Points Inside
Abstract
We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers.
Cite as
Herbert Edelsbrunner, Alexey Garber, and Morteza Saghafian. On Spheres with k Points Inside. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 43:1-43:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2025.43,
author = {Edelsbrunner, Herbert and Garber, Alexey and Saghafian, Morteza},
title = {{On Spheres with k Points Inside}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {43:1--43:12},
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.2025.43},
URN = {urn:nbn:de:0030-drops-231951},
doi = {10.4230/LIPIcs.SoCG.2025.43},
annote = {Keywords: Triangulations, higher-order Delaunay triangulations, hypertriangulations, Delone sets, k-sets, Worpitzky’s identity, hypersimplices}
}
Document
A Minor-Testing Approach for Coordinated Motion Planning with Sliding Robots
Abstract
We study a variant of the Coordinated Motion Planning problem on undirected graphs, referred to herein as the Coordinated Sliding-Motion Planning (CSMP) problem. In this variant, we are given an undirected graph G, k robots R₁,… ,R_k positioned on distinct vertices of G, p ≤ k distinct destination vertices for robots R₁,… ,R_p, and 𝓁 ∈ ℕ. The problem is to decide if there is a serial schedule of at most 𝓁 moves (i.e., of makespan 𝓁) such that at the end of the schedule each robot with a destination reaches it, where a robot’s move is a free path (unoccupied by any robots) from its current position to an unoccupied vertex. The problem is known to be NP-hard even on full grids. It has been studied in several contexts, including coin movement and reconfiguration problems, with respect to feasibility, complexity, and approximation. Geometric variants of the problem, in which congruent geometric-shape robots (e.g., unit disk/squares) slide or translate in the Euclidean plane, have also been studied extensively.
We investigate the parameterized complexity of CSMP with respect to two parameters: the number k of robots and the makespan 𝓁. As our first result, we present a fixed-parameter algorithm for CSMP parameterized by k. For our second result, we present a fixed-parameter algorithm parameterized by 𝓁 for the special case of CSMP in which only a single robot has a destination and the graph is planar. A crucial new ingredient for both of our results is that the solution admits a succinct representation as a small labeled topological minor of the input graph.
Cite as
Eduard Eiben, Robert Ganian, Iyad Kanj, and M. S. Ramanujan. A Minor-Testing Approach for Coordinated Motion Planning with Sliding Robots. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{eiben_et_al:LIPIcs.SoCG.2025.44,
author = {Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Ramanujan, M. S.},
title = {{A Minor-Testing Approach for Coordinated Motion Planning with Sliding Robots}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {44:1--44:15},
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.2025.44},
URN = {urn:nbn:de:0030-drops-231966},
doi = {10.4230/LIPIcs.SoCG.2025.44},
annote = {Keywords: coordinated motion planning on graphs, parameterized complexity, topological minor testing, planar graphs}
}
Document
Higher Order Bipartiteness vs Bi-Partitioning in Simplicial Complexes
Abstract
Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the construction of complex networks. Another key identifying feature is their Laplacian spectrum as bipartite graphs achieve the maximum possible eigenvalue of graph Laplacian. However, for modeling higher-order connections in complex systems, hypergraphs and simplicial complexes are required due to the limitations of graphs in representing pairwise interactions. In this article, using simple tools from graph theory, we extend the cycle-based characterization from bipartite graphs to those simplicial complexes that achieve the maximum Hodge Laplacian eigenvalue, known as disorientable simplicial complexes. We show that a N-dimensional simplicial complex is disorientable if its down dual graph contains no simple odd cycle of distinct edges and no twisted even cycle of distinct edges. Furthermore, we see that in a N-simplicial complex without twisting cycles, the fewer the number of (non-branching) simple odd cycles in its down dual graph, the closer is its maximum eigenvalue to the possible maximum eigenvalue of Hodge Laplacian. Similar to the graph case, the absence of odd cycles plays a crucial role in solving the bi-partitioning problem of simplexes in higher dimensions.
Cite as
Marzieh Eidi and Sayan Mukherjee. Higher Order Bipartiteness vs Bi-Partitioning in Simplicial Complexes. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 45:1-45:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{eidi_et_al:LIPIcs.SoCG.2025.45,
author = {Eidi, Marzieh and Mukherjee, Sayan},
title = {{Higher Order Bipartiteness vs Bi-Partitioning in Simplicial Complexes}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {45:1--45:12},
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.2025.45},
URN = {urn:nbn:de:0030-drops-231972},
doi = {10.4230/LIPIcs.SoCG.2025.45},
annote = {Keywords: Bipartite graphs, Simplicial complex, Disorientability, Hodge Laplacian, odd cycles, Twisted cycles, down-dual graph}
}
Document
Non-Euclidean Erdős-Anning Theorems
Abstract
The Erdős-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter δ, at most O(δ²) points can have integer distances from all three triangle vertices. We prove the same results for any strictly convex distance function on the plane, and analogous results for every two-dimensional complete Riemannian manifold of bounded genus and for geodesic distance on the boundary of every three-dimensional Euclidean convex set. As a consequence, we resolve a 1983 question of Richard Guy on the equilateral dimension of Riemannian manifolds. Our proofs are based on the properties of additively weighted Voronoi diagrams of these distances.
Cite as
David Eppstein. Non-Euclidean Erdős-Anning Theorems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{eppstein:LIPIcs.SoCG.2025.46,
author = {Eppstein, David},
title = {{Non-Euclidean Erd\H{o}s-Anning Theorems}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {46:1--46:15},
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.2025.46},
URN = {urn:nbn:de:0030-drops-231983},
doi = {10.4230/LIPIcs.SoCG.2025.46},
annote = {Keywords: integer distances, additively weighted Voronoi diagrams, convex distance functions, Riemannian manifolds, geodesic distance}
}
Document
Shelling and Sinking Graphs on the Sphere
Abstract
We describe a promising approach to efficiently morph spherical graphs, extending earlier approaches of Awartani and Henderson [Trans. AMS 1987] and Kobourov and Landis [JGAA 2006]. Specifically, we describe two methods to morph shortest-path triangulations of the sphere by moving their vertices along longitudes into the southern hemisphere; we call a triangulation sinkable if such a morph exists. Our first method generalizes a longitudinal shelling construction of Awartani and Henderson; a triangulation is sinkable if a specific orientation of its dual graph is acyclic. We describe a simple polynomial-time algorithm to find a longitudinally shellable rotation of a given spherical triangulation, if one exists; we also construct a spherical triangulation that has no longitudinally shellable rotation. Our second method is based on a linear-programming characterization of sinkability. By identifying its optimal basis, we show that this linear program can be solved in O(n^{ω/2}) time, where ω is the matrix-multiplication exponent, assuming the underlying linear system is non-singular. Finally, we pose several conjectures and describe experimental results that support them.
Cite as
Jeff Erickson and Christian Howard. Shelling and Sinking Graphs on the Sphere. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 47:1-47:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{erickson_et_al:LIPIcs.SoCG.2025.47,
author = {Erickson, Jeff and Howard, Christian},
title = {{Shelling and Sinking Graphs on the Sphere}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {47:1--47:18},
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.2025.47},
URN = {urn:nbn:de:0030-drops-231996},
doi = {10.4230/LIPIcs.SoCG.2025.47},
annote = {Keywords: morphing, planar graphs, spherical graph drawing, longitudinal shelling}
}
Document
Exact Algorithms for Minimum Dilation Triangulation
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
On Sparse Covers of Minor Free Graphs, Low Dimensional Metric Embeddings, and Other Applications
Abstract
Given a metric space (X,d_X), a (β,s,Δ)-sparse cover is a collection of clusters 𝒞 ⊆ P(X) with diameter at most Δ, such that for every point x ∈ X, the ball B_X(x,Δ/β) is fully contained in some cluster C ∈ 𝒞, and x belongs to at most s clusters in 𝒞. Our main contribution is to show that the shortest path metric of every K_r-minor free graphs admits (O(r),O(r²),Δ)-sparse cover, and for every ε > 0, (4+ε,O(1/ε)^r,Δ)-sparse cover (for arbitrary Δ > 0). We then use this sparse cover to show that every K_r-minor free graph embeds into 𝓁_∞^{Õ(1/ε)^{r+1}⋅log n} with distortion 3+ε (resp. into 𝓁_∞^{Õ(r²)⋅log n} with distortion O(r)). Further, among other applications, this sparse cover immediately implies an algorithm for the oblivious buy-at-bulk problem in fixed minor free graphs with the tight approximation factor O(log n) (previously nothing beyond general graphs was known).
Cite as
Arnold Filtser. On Sparse Covers of Minor Free Graphs, Low Dimensional Metric Embeddings, and Other Applications. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 49:1-49:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{filtser:LIPIcs.SoCG.2025.49,
author = {Filtser, Arnold},
title = {{On Sparse Covers of Minor Free Graphs, Low Dimensional Metric Embeddings, and Other Applications}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {49:1--49:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.49},
URN = {urn:nbn:de:0030-drops-232015},
doi = {10.4230/LIPIcs.SoCG.2025.49},
annote = {Keywords: Sparse cover, minor free graphs, metric embeddings, 𝓁\underline∞, oblivious buy-at-bulk}
}
Document
Immersions and Albertson’s Conjecture
Abstract
A graph is said to contain K_k (a clique of size k) as a weak immersion if it has k vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger’s conjecture: Every graph of chromatic number k contains K_k as a weak immersion. We prove this conjecture for graphs with at most 1.4(k-1) vertices. As an application, we make some progress on Albertson’s conjecture on crossing numbers of graphs, according to which every graph G with chromatic number k satisfies cr(G) ≥ cr(K_k). In particular, we show that the conjecture is true for all graphs of chromatic number k, provided that they have at most 1.4(k-1) vertices and k is sufficiently large.
Cite as
Jacob Fox, János Pach, and Andrew Suk. Immersions and Albertson’s Conjecture. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 50:1-50:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{fox_et_al:LIPIcs.SoCG.2025.50,
author = {Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew},
title = {{Immersions and Albertson’s Conjecture}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {50:1--50:10},
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.2025.50},
URN = {urn:nbn:de:0030-drops-232022},
doi = {10.4230/LIPIcs.SoCG.2025.50},
annote = {Keywords: Immersions, crossing numbers, chromatic number}
}
Document
Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction
Abstract
We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the essential computation of the combine step of a quantum divide-and-conquer algorithm prior to the conquer step while avoiding recursion. This model is better suited for the quantum setting, due to its non-recursive nature. We show the utility of this approach by providing quantum algorithms for 2D maxima set and convex hull problems for sorted point sets running in Õ(√{nh}) time, w.h.p., where h is the size of the output.
Cite as
Shion Fukuzawa, Michael T. Goodrich, and Sandy Irani. Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{fukuzawa_et_al:LIPIcs.SoCG.2025.51,
author = {Fukuzawa, Shion and Goodrich, Michael T. and Irani, Sandy},
title = {{Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {51:1--51:15},
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.2025.51},
URN = {urn:nbn:de:0030-drops-232035},
doi = {10.4230/LIPIcs.SoCG.2025.51},
annote = {Keywords: quantum computing, computational geometry, divide and conquer, convex hulls, maxima sets}
}
Document
Uniform Bounds on Product Sylvester-Gallai Configurations
Abstract
In this work, we explore a non-linear extension of the classical Sylvester-Gallai configuration. Let 𝕂 be an algebraically closed field of characteristic zero, and let ℱ = {F_1, …, F_m} ⊂ 𝕂[x_1, …, x_N] denote a collection of irreducible homogeneous polynomials of degree at most d, where each F_i is not a scalar multiple of any other F_j for i ≠ j. We define ℱ to be a product Sylvester-Gallai configuration if, for any two distinct polynomials F_i, F_j ∈ ℱ, the following condition is satisfied: ∏_{k≠i, j} F_k ∈ rad (F_i, F_j) .
We prove that product Sylvester-Gallai configurations are inherently low dimensional. Specifically, we show that there exists a function λ : ℕ → ℕ, independent of 𝕂, N, and m, such that any product Sylvester-Gallai configuration must satisfy: dim(span_𝕂(ℱ)) ≤ λ(d).
This result generalizes the main theorems from (Shpilka 2019, Peleg and Shpilka 2020, Oliveira and Sengupta 2023), and gets us one step closer to a full derandomization of the polynomial identity testing problem for the class of depth 4 circuits with bounded top and bottom fan-in.
Cite as
Abhibhav Garg, Rafael Oliveira, and Akash Kumar Sengupta. Uniform Bounds on Product Sylvester-Gallai Configurations. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{garg_et_al:LIPIcs.SoCG.2025.52,
author = {Garg, Abhibhav and Oliveira, Rafael and Sengupta, Akash Kumar},
title = {{Uniform Bounds on Product Sylvester-Gallai Configurations}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {52:1--52:15},
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.2025.52},
URN = {urn:nbn:de:0030-drops-232043},
doi = {10.4230/LIPIcs.SoCG.2025.52},
annote = {Keywords: Sylvester-Gallai theorem, arrangements of hypersurfaces, algebraic complexity, polynomial identity testing, algebraic geometry, commutative algebra}
}
Document
A PTAS for TSP with Neighbourhoods over Parallel Line Segments
Abstract
We consider the Travelling Salesman Problem with Neighbourhoods (TSPN) on the Euclidean plane (ℝ²) and present a Polynomial-Time Approximation Scheme (PTAS) when the neighbourhoods are parallel line segments with lengths between [1, λ] for any constant value λ ≥ 1. In TSPN (which generalizes classic TSP), each client represents a set (or neighbourhood) of points in a metric and the goal is to find a minimum cost TSP tour that visits at least one point from each client set. In the Euclidean setting, each neighbourhood is a region on the plane. TSPN is significantly more difficult than classic TSP even in the Euclidean setting, as it captures group TSP. A notable case of TSPN is when each neighbourhood is a line segment. Although there are PTASs for when neighbourhoods are fat objects (with limited overlap), TSPN over line segments is APX-hard even if all the line segments have unit length. For parallel (unit) line segments, the best approximation factor is 3√2 from more than two decades ago. The PTAS we present in this paper settles the approximability of this case of the problem. Our algorithm finds a (1 + ε)-factor approximation for an instance of the problem for n segments with lengths in [1,λ] in time n^O(λ/ε³).
Cite as
Benyamin Ghaseminia and Mohammad R. Salavatipour. A PTAS for TSP with Neighbourhoods over Parallel Line Segments. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ghaseminia_et_al:LIPIcs.SoCG.2025.53,
author = {Ghaseminia, Benyamin and Salavatipour, Mohammad R.},
title = {{A PTAS for TSP with Neighbourhoods over Parallel Line Segments}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {53:1--53:15},
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.2025.53},
URN = {urn:nbn:de:0030-drops-232058},
doi = {10.4230/LIPIcs.SoCG.2025.53},
annote = {Keywords: Approximation Scheme, TSP Neighbourhood, Parallel line segments}
}
Document
The Fréchet Distance Unleashed: Approximating a Dog with a Frog
Abstract
We show that a variant of the continuous Fréchet distance between polygonal curves can be computed using essentially the same algorithm used to solve the discrete version. The new variant is not necessarily monotone, but this shortcoming can be easily handled via refinement.
Combined with a Dijkstra/Prim type algorithm, this leads to a realization of the Fréchet distance (i.e., a morphing) that is locally optimal (aka locally correct), that is both easy to compute, and in practice, takes near linear time on many inputs. The new morphing has the property that the leash is always as short as possible. These matchings/morphings are more natural, and are better than the ones computed by standard algorithms - in particular, they handle noise more graciously. This should make the Fréchet distance more useful for real world applications.
We implemented the new algorithm, and various strategies to obtain fast practical performance. We performed extensive experiments with our new algorithm, and released publicly available (and easily installable and usable) Julia and Python packages. In particular, the Julia implementation, for computing the regular Fréchet distance, seems to be {significantly faster} than other currently available implementations. See Table 2.2. Our algorithms can be used to compute the almost-exact Fréchet distance between polygonal curves.
Implementations and numerous examples are available here: https://frechet.xyz.
Cite as
Sariel Har-Peled, Benjamin Raichel, and Eliot W. Robson. The Fréchet Distance Unleashed: Approximating a Dog with a Frog. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{harpeled_et_al:LIPIcs.SoCG.2025.54,
author = {Har-Peled, Sariel and Raichel, Benjamin and Robson, Eliot W.},
title = {{The Fr\'{e}chet Distance Unleashed: Approximating a Dog with a Frog}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {54:1--54:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.54},
URN = {urn:nbn:de:0030-drops-232066},
doi = {10.4230/LIPIcs.SoCG.2025.54},
annote = {Keywords: Curve similarity, Fr\'{e}chet distance}
}
Document
A Practical Algorithm for Knot Factorisation
Abstract
We present an algorithm for computing the prime factorisation of a knot, which is practical in the following sense: using Regina, we give an implementation that works well for inputs of reasonable size, including prime knots from the 19-crossing census. The main new ingredient in this work is an object that we call an "edge-ideal triangulation", which is what our algorithm uses to represent knots. As other applications, we give an alternative proof that prime knot recognition is in coNP, and present some new complexity results for triangulations. Beyond knots, our work showcases edge-ideal triangulations as a tool for potential applications in 3-manifold topology.
Cite as
Alexander He, Eric Sedgwick, and Jonathan Spreer. A Practical Algorithm for Knot Factorisation. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{he_et_al:LIPIcs.SoCG.2025.55,
author = {He, Alexander and Sedgwick, Eric and Spreer, Jonathan},
title = {{A Practical Algorithm for Knot Factorisation}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {55:1--55:15},
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.2025.55},
URN = {urn:nbn:de:0030-drops-232075},
doi = {10.4230/LIPIcs.SoCG.2025.55},
annote = {Keywords: Prime and composite knots, (crushing) normal surfaces, edge-ideal triangulations, co-NP certificate, triangulation complexity}
}
Document
Sublinear Data Structures for Nearest Neighbor in Ultra High Dimensions
Abstract
Geometric data structures have been extensively studied in the regime where the dimension is much smaller than the number of input points. But in many scenarios in Machine Learning, the dimension can be much higher than the number of points and can be so high that the data structure might be unable to read and store all coordinates of the input and query points.
Inspired by these scenarios and related studies in feature selection and explainable clustering, we initiate the study of geometric data structures in this ultra-high dimensional regime. Our focus is the approximate nearest neighbor problem.
In this problem, we are given a set of n points C ⊆ ℝ^d and have to produce a small data structure that can quickly answer the following query: given q ∈ ℝ^d, return a point c ∈ C that is approximately nearest to q, where the distance is under 𝓁₁, 𝓁₂, or other norms. Many groundbreaking (1+ε)-approximation algorithms have recently been discovered for 𝓁₁- and 𝓁₂-norm distances in the regime where d≪ n.
The main question in this paper is: Is there a data structure with sublinear (o(nd)) space and sublinear (o(d)) query time when d≫ n? This question can be partially answered from the machine-learning literature:
- For 𝓁₁-norm distances, an Õ(log(n))-approximation data structure with Õ(n log d) space and O(n) query time can be obtained from explainable clustering techniques [Dasgupta et al. ICML'20; Makarychev and Shan ICML'21; Esfandiari, Mirrokni, and Narayanan SODA'22; Gamlath et al. NeurIPS'21; Charikar and Hu SODA'22].
- For 𝓁₂-norm distances, a (√3+ε)-approximation data structure with Õ(n log(d)/poly(ε)) space and Õ(n/poly(ε)) query time can be obtained from feature selection techniques [Boutsidis, Drineas, and Mahoney NeurIPS'09; Boutsidis et al. IEEE Trans. Inf. Theory'15; Cohen et al. STOC'15].
- For 𝓁_p-norm distances, a O(n^{p-1}log²(n))-approximation data structure with O(nlog(n) + nlog(d)) space and O(n) query time can be obtained from the explainable clustering algorithms of [Gamlath et al. NeurIPS'21].
An important open problem is whether a (1+ε)-approximation data structure exists. This is not known for any norm, even with higher (e.g. poly(n)⋅ o(d)) space and query time.
In this paper, we answer this question affirmatively. We present (1+ε)-approximation data structures with the following guarantees.
- For 𝓁₁- and 𝓁₂-norm distances: Õ(n log(d)/poly(ε)) space and Õ(n/poly(ε)) query time. We show that these space and time bounds are tight up to poly (log n/ε) factors.
- For 𝓁_p-norm distances: Õ(n² log(d) (log log(n)/ε)^p) space and Õ (n(log log(n)/ε)^p) query time.
Via simple reductions, our data structures imply sublinear-in-d data structures for some other geometric problems; e.g. approximate orthogonal range search (in the style of [Arya and Mount SoCG'95]), furthest neighbor, and give rise to a sublinear O(1)-approximate representation of k-median and k-means clustering. We hope that this paper inspires future work on sublinear geometric data structures.
Cite as
Martin G. Herold, Danupon Nanongkai, Joachim Spoerhase, Nithin Varma, and Zihang Wu. Sublinear Data Structures for Nearest Neighbor in Ultra High Dimensions. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{herold_et_al:LIPIcs.SoCG.2025.56,
author = {Herold, Martin G. and Nanongkai, Danupon and Spoerhase, Joachim and Varma, Nithin and Wu, Zihang},
title = {{Sublinear Data Structures for Nearest Neighbor in Ultra High Dimensions}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {56:1--56:15},
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.2025.56},
URN = {urn:nbn:de:0030-drops-232087},
doi = {10.4230/LIPIcs.SoCG.2025.56},
annote = {Keywords: sublinear data structure, approximate nearest neighbor}
}
Document
Snap Rounding: A Cautionary Tale
Abstract
This paper describes the author’s experience using and modifying the technique of snap rounding to meet the needs of a circuit verification system. The interplay of theory and practice illuminates the subtle challenges of both.
Cite as
John Hershberger. Snap Rounding: A Cautionary Tale. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 57:1-57:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{hershberger:LIPIcs.SoCG.2025.57,
author = {Hershberger, John},
title = {{Snap Rounding: A Cautionary Tale}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {57:1--57:14},
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.2025.57},
URN = {urn:nbn:de:0030-drops-232097},
doi = {10.4230/LIPIcs.SoCG.2025.57},
annote = {Keywords: Snap rounding, implementation, computational geometry}
}
Document
Tracking the Persistence of Harmonic Chains: Barcode and Stability
Abstract
The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of data, the persistence barcode tracks the evolution of its homology groups. In this paper, we introduce a new type of barcode, called the harmonic chain barcode, which tracks the evolution of harmonic chains. In addition, we show that the harmonic chain barcode is stable. Given a filtration of a simplicial complex of size m, we present an algorithm to compute its harmonic chain barcode in O(m³) time. Consequently, the harmonic chain barcode can enrich the family of topological descriptors in applications where a persistence barcode is applicable, such as feature vectorization and machine learning.
Cite as
Tao Hou, Salman Parsa, and Bei Wang. Tracking the Persistence of Harmonic Chains: Barcode and Stability. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{hou_et_al:LIPIcs.SoCG.2025.58,
author = {Hou, Tao and Parsa, Salman and Wang, Bei},
title = {{Tracking the Persistence of Harmonic Chains: Barcode and Stability}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {58:1--58:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.58},
URN = {urn:nbn:de:0030-drops-232100},
doi = {10.4230/LIPIcs.SoCG.2025.58},
annote = {Keywords: Persistent homology, harmonic chains, topological data analysis}
}
Document
Incremental Planar Nearest Neighbor Queries with Optimal Query Time
Abstract
In this paper we show that two-dimensional nearest neighbor queries can be answered in optimal O(log n) time while supporting insertions in O(log^{1+ε} n) time. No previous data structure was known that supports O(log n)-time queries and polylog-time insertions. In order to achieve logarithmic queries our data structure uses a new technique related to fractional cascading that leverages the inherent geometry of this problem. Our method can be also used in other semi-dynamic scenarios.
Cite as
John Iacono and Yakov Nekrich. Incremental Planar Nearest Neighbor Queries with Optimal Query Time. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{iacono_et_al:LIPIcs.SoCG.2025.59,
author = {Iacono, John and Nekrich, Yakov},
title = {{Incremental Planar Nearest Neighbor Queries with Optimal Query Time}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {59:1--59:15},
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.2025.59},
URN = {urn:nbn:de:0030-drops-232117},
doi = {10.4230/LIPIcs.SoCG.2025.59},
annote = {Keywords: Data Structures, Dynamic Data Structures, Nearest Neighbor Queries}
}
Document
Improved Approximation Algorithms for Three-Dimensional Knapsack
Abstract
We study the three-dimensional Knapsack (3DK) problem, in which we are given a set of axis-aligned cuboids with associated profits and an axis-aligned cube knapsack. The objective is to find a non-overlapping axis-aligned packing (by translation) of the maximum profit subset of cuboids into the cube. The previous best approximation algorithm is due to Diedrich, Harren, Jansen, Thöle, and Thomas (2008), who gave a (7+ε)-approximation algorithm for 3DK and a (5+ε)-approximation algorithm for the variant when the items can be rotated by 90 degrees around any axis, for any constant ε > 0. Chlebík and Chlebíková (2009) showed that the problem does not admit an asymptotic polynomial-time approximation scheme.
We provide an improved polynomial-time (139/29+ε) ≈ 4.794-approximation algorithm for 3DK and (30/7+ε) ≈ 4.286-approximation when rotations by 90 degrees are allowed. We also provide improved approximation algorithms for several variants such as the cardinality case (when all items have the same profit) and uniform profit-density case (when the profit of an item is equal to its volume). Our key technical contribution is container packing - a structured packing in 3D such that all items are assigned into a constant number of containers, and each container is packed using a specific strategy based on its type. We first show the existence of highly profitable container packings. Thereafter, we show that one can find near-optimal container packing efficiently using a variant of the Generalized Assignment Problem (GAP).
Cite as
Klaus Jansen, Debajyoti Kar, Arindam Khan, K. V. N. Sreenivas, and Malte Tutas. Improved Approximation Algorithms for Three-Dimensional Knapsack. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 60:1-60:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{jansen_et_al:LIPIcs.SoCG.2025.60,
author = {Jansen, Klaus and Kar, Debajyoti and Khan, Arindam and Sreenivas, K. V. N. and Tutas, Malte},
title = {{Improved Approximation Algorithms for Three-Dimensional Knapsack}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {60:1--60:18},
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.2025.60},
URN = {urn:nbn:de:0030-drops-232126},
doi = {10.4230/LIPIcs.SoCG.2025.60},
annote = {Keywords: Approximation Algorithms, Hyperrectangle Packing, Multidimensional Knapsack, Three-dimensional Packing}
}
Document
k-Dimensional Transversals for Fat Convex Sets
Abstract
We prove a fractional Helly theorem for k-flats intersecting fat convex sets. A family ℱ of sets is said to be ρ-fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by ρ. We prove that for every dimension d and positive reals ρ and α there exists a positive β = β(d,ρ, α) such that if ℱ is a finite family of ρ-fat convex sets in ℝ^d and an α-fraction of the (k+2)-size subfamilies from ℱ can be hit by a k-flat, then there is a k-flat that intersects at least a β-fraction of the sets of ℱ. We prove spherical and colorful variants of the above results and prove a (p,k+2)-theorem for k-flats intersecting balls.
Cite as
Attila Jung and Dömötör Pálvölgyi. k-Dimensional Transversals for Fat Convex Sets. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 61:1-61:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{jung_et_al:LIPIcs.SoCG.2025.61,
author = {Jung, Attila and P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r},
title = {{k-Dimensional Transversals for Fat Convex Sets}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {61:1--61:12},
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.2025.61},
URN = {urn:nbn:de:0030-drops-232136},
doi = {10.4230/LIPIcs.SoCG.2025.61},
annote = {Keywords: discrete geometry, transversals, Helly, hypergraphs}
}
Document
On Approximability of 𝓁₂² Min-Sum Clustering
Abstract
The 𝓁₂² min-sum k-clustering problem is to partition an input set into clusters C_1,…,C_k to minimize ∑_{i=1}^k ∑_{p,q ∈ C_i} ‖p-q‖₂². Although 𝓁₂² min-sum k-clustering is NP-hard, it is not known whether it is NP-hard to approximate 𝓁₂² min-sum k-clustering beyond a certain factor.
In this paper, we give the first hardness-of-approximation result for the 𝓁₂² min-sum k-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than 1.056 and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327.
We then complement our hardness result by giving a fast PTAS for 𝓁₂² min-sum k-clustering. Specifically, our algorithm runs in time O(n^{1+o(1)}d⋅ 2^{(k/ε)^O(1)}), which is the first nearly linear time algorithm for this problem. We also consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label i ∈ [k] for input point, thereby implicitly partitioning the input dataset into k clusters that induce an approximately optimal solution, up to some amount of adversarial error α ∈ [0,1/2). We give a polynomial-time algorithm that outputs a (1+γα)/(1-α)²-approximation to 𝓁₂² min-sum k-clustering, for a fixed constant γ > 0.
Cite as
Karthik C. S., Euiwoong Lee, Yuval Rabani, Chris Schwiegelshohn, and Samson Zhou. On Approximability of 𝓁₂² Min-Sum Clustering. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 62:1-62:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{karthikc.s._et_al:LIPIcs.SoCG.2025.62,
author = {Karthik C. S. and Lee, Euiwoong and Rabani, Yuval and Schwiegelshohn, Chris and Zhou, Samson},
title = {{On Approximability of 𝓁₂² Min-Sum Clustering}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {62:1--62:18},
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.2025.62},
URN = {urn:nbn:de:0030-drops-232142},
doi = {10.4230/LIPIcs.SoCG.2025.62},
annote = {Keywords: Clustering, hardness of approximation, polynomial-time approximation schemes, learning-augmented algorithms}
}
Document
The Maximum Clique Problem in a Disk Graph Made Easy
Abstract
A disk graph is an intersection graph of disks in ℝ². Determining the computational complexity of finding a maximum clique in a disk graph is a long-standing open problem. In 1990, Clark, Colbourn, and Johnson gave a polynomial-time algorithm for computing a maximum clique in a unit disk graph. However, finding a maximum clique when disks are of arbitrary size is widely believed to be a challenging open problem. In this paper, we provide a new perspective to examine adjacencies in a disk graph that helps obtain the following results.
- We design an 𝒪^*(n^{2k})-time algorithm, where 𝒪^* hides a polynomial factor, to find a maximum clique in a n-vertex disk graph with k different sizes of radii. This is polynomial for every fixed k, and thus settles the open question for the case when k = 2.
- Given a set of n unit disks, we show how to compute a maximum clique inside each possible axis-aligned rectangle determined by the disk centers in O(n⁵log n)-time. This is at least a factor of n^{4/3} faster than applying the fastest known algorithm for finding a maximum clique in a unit disk graph for each rectangle independently.
- We give an 𝒪^*(n^{2rk})-time algorithm to find a maximum clique in a n-vertex ball graph with k different sizes of radii where the ball centers lie on r parallel planes. This is polynomial for every fixed k and r, and thus contrasts the previously known NP-hardness result for finding a maximum clique in an arbitrary ball graph.
- We design an 𝒪^*(n^{2k})-time algorithm to find a maximum clique in the intersection graph of a set S of n L-visible convex polygons, where k is the number of distinct shapes in S. This contrasts the known hardness result on finding a maximum clique in the intersection graph of unit disks and axis-aligned rectangles.
Cite as
J. Mark Keil and Debajyoti Mondal. The Maximum Clique Problem in a Disk Graph Made Easy. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 63:1-63:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{keil_et_al:LIPIcs.SoCG.2025.63,
author = {Keil, J. Mark and Mondal, Debajyoti},
title = {{The Maximum Clique Problem in a Disk Graph Made Easy}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {63:1--63:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.63},
URN = {urn:nbn:de:0030-drops-232155},
doi = {10.4230/LIPIcs.SoCG.2025.63},
annote = {Keywords: Geometric Intersection Graphs, Disk Graphs, Ball Graphs, Maximum Clique}
}
Document
Super-Polynomial Growth of the Generalized Persistence Diagram
Abstract
The Generalized Persistence Diagram (GPD) for multi-parameter persistence naturally extends the classical notion of persistence diagram for one-parameter persistence. However, unlike its classical counterpart, computing the GPD remains a significant challenge. The main hurdle is that, while the GPD is defined as the Möbius inversion of the Generalized Rank Invariant (GRI), computing the GRI is intractable due to the formidable size of its domain, i.e., the set of all connected and convex subsets in a finite grid in ℝ^d with d ≥ 2. This computational intractability suggests seeking alternative approaches to computing the GPD.
In order to study the complexity associated to computing the GPD, it is useful to consider its classical one-parameter counterpart, where for a filtration of a simplicial complex with n simplices, its persistence diagram contains at most n points. This observation leads to the question: Given a d-parameter simplicial filtration, could the cardinality of its GPD (specifically, the support of the GPD) also be bounded by a polynomial in the number of simplices in the filtration? This is the case for d = 1, where we compute the persistence diagram directly at the simplicial filtration level. If this were also the case for d ≥ 2, it might be possible to compute the GPD directly and much more efficiently without relying on the GRI.
We show that the answer to the question above is negative, demonstrating the inherent difficulty of computing the GPD. More specifically, we construct a sequence of d-parameter simplicial filtrations where the cardinalities of their GPDs are not bounded by any polynomial in the number of simplices. Furthermore, we show that several commonly used methods for constructing multi-parameter filtrations can give rise to such "wild" filtrations.
Cite as
Donghan Kim, Woojin Kim, and Wonjun Lee. Super-Polynomial Growth of the Generalized Persistence Diagram. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 64:1-64:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kim_et_al:LIPIcs.SoCG.2025.64,
author = {Kim, Donghan and Kim, Woojin and Lee, Wonjun},
title = {{Super-Polynomial Growth of the Generalized Persistence Diagram}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {64:1--64:20},
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.2025.64},
URN = {urn:nbn:de:0030-drops-232162},
doi = {10.4230/LIPIcs.SoCG.2025.64},
annote = {Keywords: Persistent homology, M\"{o}bius inversion, Multiparameter persistence, Generalized persistence diagram, Generalized rank invariant}
}
Document
Recognizing 2-Layer and Outer k-Planar Graphs
Abstract
The crossing number of a graph is the least number of crossings over all drawings of the graph in the plane. Computing the crossing number of a given graph is NP-hard, but fixed-parameter tractable (FPT) with respect to the natural parameter. Two well-known variants of the problem are 2-layer crossing minimization and circular crossing minimization, where every vertex must lie on one of two layers, namely two parallel lines, or a circle, respectively. In both cases, edges are drawn as straight-line segments. Both variants are NP-hard, but admit FPT-algorithms with respect to the natural parameter.
In recent years, in the context of beyond-planar graphs, a local version of the crossing number has also received considerable attention. A graph is k-planar if it admits a drawing with at most k crossings per edge. In contrast to the crossing number, recognizing k-planar graphs is NP-hard even if k = 1 and hence not likely to be FPT with respect to the natural parameter k.
In this paper, we consider the two above variants in the local setting. The k-planar graphs that admit a straight-line drawing with vertices on two layers or on a circle are called 2-layer k-planar and outer k-planar graphs, respectively. We study the parameterized complexity of the two recognition problems with respect to the natural parameter k. For k = 0, the two classes of graphs are exactly the caterpillars and outerplanar graphs, respectively, which can be recognized in linear time. Two groups of researchers independently showed that outer 1-planar graphs can also be recognized in linear time [Hong et al., Algorithmica 2015; Auer et al., Algorithmica 2016]. One group asked explicitly whether outer 2-planar graphs can be recognized in polynomial time.
Our main contribution consists of XP-algorithms for recognizing 2-layer k-planar graphs and outer k-planar graphs, which implies that both recognition problems can be solved in polynomial time for every fixed k. We complement these results by showing that recognizing 2-layer k-planar graphs is XNLP-complete and that recognizing outer k-planar graphs is XNLP-hard. This implies that both problems are W[t]-hard for every t and that it is unlikely that they admit FPT-algorithms. On the other hand, we present an FPT-algorithm for recognizing 2-layer k-planar graphs where the order of the vertices on one layer is specified.
Cite as
Yasuaki Kobayashi, Yuto Okada, and Alexander Wolff. Recognizing 2-Layer and Outer k-Planar Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 65:1-65:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kobayashi_et_al:LIPIcs.SoCG.2025.65,
author = {Kobayashi, Yasuaki and Okada, Yuto and Wolff, Alexander},
title = {{Recognizing 2-Layer and Outer k-Planar Graphs}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {65:1--65:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.65},
URN = {urn:nbn:de:0030-drops-232170},
doi = {10.4230/LIPIcs.SoCG.2025.65},
annote = {Keywords: 2-layer k-planar graphs, outer k-planar graphs, recognition algorithms, local crossing number, bandwidth, FPT, XNLP, XP, W\lbrackt\rbrack}
}
Document
Lipschitz Decompositions of Finite 𝓁_{p} Metrics
Abstract
Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for n-point subsets of 𝓁_p, for p > 2, remained open, see e.g. [Naor, SODA 2017]. We make significant progress on this question and establish the bound β = O(log^{1-1/p} n). Building on prior work, we demonstrate applications of this result to two problems, high-dimensional geometric spanners and distance labeling schemes. In addition, we sharpen a related decomposition bound for 1 < p < 2, due to Filtser and Neiman [Algorithmica 2022].
Cite as
Robert Krauthgamer and Nir Petruschka. Lipschitz Decompositions of Finite 𝓁_{p} Metrics. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{krauthgamer_et_al:LIPIcs.SoCG.2025.66,
author = {Krauthgamer, Robert and Petruschka, Nir},
title = {{Lipschitz Decompositions of Finite 𝓁\underline\{p\} Metrics}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {66:1--66:14},
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.2025.66},
URN = {urn:nbn:de:0030-drops-232182},
doi = {10.4230/LIPIcs.SoCG.2025.66},
annote = {Keywords: Lipschitz decompositions, metric embeddings, geometric spanners}
}
Document
Hard Diagrams of Split Links
Abstract
Deformations of knots and links in ambient space can be studied combinatorially on their diagrams via local modifications called Reidemeister moves. While it is well-known that, in order to move between equivalent diagrams with Reidemeister moves, one sometimes needs to insert excess crossings, there are significant gaps between the best known lower and upper bounds on the required number of these added crossings. In this article, we study the problem of turning a diagram of a split link into a split diagram, and we show that there exist split links with diagrams requiring an arbitrarily large number of such additional crossings. More precisely, we provide a family of diagrams of split links, so that any sequence of Reidemeister moves transforming a diagram with c crossings into a split diagram requires going through a diagram with Ω(√c) extra crossings. Our proof relies on the framework of bubble tangles, as introduced by the first two authors, and a technique of Chambers and Liokumovitch to turn homotopies into isotopies in the context of Riemannian geometry.
Cite as
Corentin Lunel, Arnaud de Mesmay, and Jonathan Spreer. Hard Diagrams of Split Links. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 67:1-67:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{lunel_et_al:LIPIcs.SoCG.2025.67,
author = {Lunel, Corentin and de Mesmay, Arnaud and Spreer, Jonathan},
title = {{Hard Diagrams of Split Links}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {67:1--67:17},
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.2025.67},
URN = {urn:nbn:de:0030-drops-232191},
doi = {10.4230/LIPIcs.SoCG.2025.67},
annote = {Keywords: Knot theory, hard knot and link diagrams, Reidemeister moves, extra crossings, split links, bubble tangles, compression representativity}
}
Document
Persistent (Co)Homology in Matrix Multiplication Time
Abstract
Most algorithms for computing persistent homology do so by tracking cycles that represent homology classes. There are many choices of such cycles, and specific choices have found different uses in applications. Although it is known that persistence diagrams can be computed in matrix multiplication time for the more general case of zigzag persistent homology [Milosavljević et al., 2011], it is not clear how to extract cycle representatives, especially if specific representatives are desired. In this paper, we provide the same matrix multiplication bound for computing representatives for the two choices common in applications in the case of ordinary persistent (co)homology. We first provide a fast version of the reduction algorithm, which is simpler than the algorithm in [Milosavljević et al., 2011], but returns a different set of representatives than the standard algorithm [Edelsbrunner et al., 2002]. We then give a fast version of a variant called the row algorithm [De Silva et al., 2011], which returns the same representatives as the standard algorithm.
Cite as
Dmitriy Morozov and Primoz Skraba. Persistent (Co)Homology in Matrix Multiplication Time. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 68:1-68:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{morozov_et_al:LIPIcs.SoCG.2025.68,
author = {Morozov, Dmitriy and Skraba, Primoz},
title = {{Persistent (Co)Homology in Matrix Multiplication Time}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {68:1--68:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.68},
URN = {urn:nbn:de:0030-drops-232204},
doi = {10.4230/LIPIcs.SoCG.2025.68},
annote = {Keywords: persistent homology, matrix multiplication, cycle representatives}
}
Document
Computing Betti Tables and Minimal Presentations of Zero-Dimensional Persistent Homology
Abstract
The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and applied mathematics, and in particular in topological data analysis, where they are known as persistence modules, and where their Betti tables describe the places at which the homology of filtered simplicial complexes changes. Although Betti tables of singly and bigraded modules are already being used in applications of topological data analysis, their computation in the bigraded case (which relies on an algorithm that is cubic in the size of the filtered simplicial complex) is a bottleneck when working with large datasets. We show that, in the special case of 0-dimensional homology (relevant for clustering and graph classification) Betti tables of bigraded modules can be computed in log-linear time. We also consider the problem of computing minimal presentations, and show that minimal presentations of 0-dimensional persistent homology can be computed in quadratic time, regardless of the grading poset.
Cite as
Dmitriy Morozov and Luis Scoccola. Computing Betti Tables and Minimal Presentations of Zero-Dimensional Persistent Homology. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 69:1-69:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{morozov_et_al:LIPIcs.SoCG.2025.69,
author = {Morozov, Dmitriy and Scoccola, Luis},
title = {{Computing Betti Tables and Minimal Presentations of Zero-Dimensional Persistent Homology}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {69:1--69:15},
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.2025.69},
URN = {urn:nbn:de:0030-drops-232219},
doi = {10.4230/LIPIcs.SoCG.2025.69},
annote = {Keywords: Multiparameter persistence, Zero-dimensional homology, Minimal presentation, Betti table}
}
Document
Computing Geomorphologically Salient Networks via Discrete Morse Theory
Abstract
Rivers, estuaries, intertidal zones, and other hydrological systems often give rise to complex networks of interconnected channels. Even today, such networks are typically drawn manually by domain experts. Traditional watershed methods for automating this process, where water flows are assumed to follow steepest descent, fail to capture behavior particular to low-relief terrains. At SoCG 2017, Kleinhans et al. proposed a method to construct a network of source-to-sink paths separated by sufficient sediment volume. However, this method is unstable with respect to minor changes of the input terrain, and constructs only channels that flow from one side of the terrain to the other, thereby failing to detect the dead-end channels ("fingers") that characterize intertidal zones.
We show how to compute geomorphologically salient networks that avoid these issues. After extending elevation data to a discrete Morse function on the terrain, we identify channels that flow through saddles and have sufficient volume of sediment on both sides. We then detect fingers, which follow the boundary of "spurs" that have sufficient volume of sediment above a particular height. The main challenge here lies in meaningfully modeling salient spurs and determining suitable heights to measure volume. We implemented our method and applied it to real-world data. Our expert users have validated the mathematical modeling by confirming that the resulting (finger) channels indeed constitute a geomorphologically salient network.
Cite as
Tim Ophelders, Anna Schenfisch, Willem Sonke, and Bettina Speckmann. Computing Geomorphologically Salient Networks via Discrete Morse Theory. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 70:1-70:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ophelders_et_al:LIPIcs.SoCG.2025.70,
author = {Ophelders, Tim and Schenfisch, Anna and Sonke, Willem and Speckmann, Bettina},
title = {{Computing Geomorphologically Salient Networks via Discrete Morse Theory}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {70:1--70:17},
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.2025.70},
URN = {urn:nbn:de:0030-drops-232221},
doi = {10.4230/LIPIcs.SoCG.2025.70},
annote = {Keywords: hydrology, network detection, intertidal zones, braided rivers, discrete Morse theory, volume persistence}
}
Document
Banana Trees for the Persistence in Time Series Experimentally
Abstract
In numerous fields, dynamic time series data require continuous updates, necessitating efficient data processing techniques for accurate analysis. This paper examines the banana tree data structure, specifically designed to efficiently maintain the multi-scale topological descriptor commonly known as persistent homology for dynamically changing time series data. We implement this data structure and conduct an experimental study to assess its properties and runtime for update operations. Our findings indicate that banana trees are highly effective with unbiased random data, outperforming state-of-the-art static algorithms in these scenarios. Additionally, our results show that real-world time series share structural properties with unbiased random walks, suggesting potential practical utility for our implementation.
Cite as
Lara Ost, Sebastiano Cultrera di Montesano, and Herbert Edelsbrunner. Banana Trees for the Persistence in Time Series Experimentally. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 71:1-71:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ost_et_al:LIPIcs.SoCG.2025.71,
author = {Ost, Lara and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert},
title = {{Banana Trees for the Persistence in Time Series Experimentally}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {71:1--71:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.71},
URN = {urn:nbn:de:0030-drops-232237},
doi = {10.4230/LIPIcs.SoCG.2025.71},
annote = {Keywords: persistent homology, time series, data structures, computational experiments}
}
Document
Geometric Bipartite Matching Based Exact Algorithms for Server Problems
Abstract
For any given metric space, obtaining an offline optimal solution to the classical k-server problem can be reduced to solving a minimum-cost partial bipartite matching between two point sets A and B within that metric space.
For d-dimensional 𝓁_p metric space, we present an Õ(min{nk, n^{2-1/(2d+1)}log Δ}⋅ Φ(n)) time algorithm for solving this instance of minimum-cost partial bipartite matching; here, Δ represents the spread of the point set, and Φ(n) is the query/update time of a d-dimensional dynamic weighted nearest neighbor data structure. Our algorithm improves upon prior algorithms that require at least Ω(nkΦ(n)) time. The design of minimum-cost (partial) bipartite matching algorithms that make sub-quadratic queries to a weighted nearest-neighbor data structure, even for bounded spread instances, is a major open problem in computational geometry. We resolve this problem at least for the instances that are generated by the offline version of the k-server problem.
Our algorithm employs a hierarchical partitioning approach, dividing the points of A∪ B into rectangles. It maintains a partial minimum-cost matching where any point b ∈ B is either matched to another point a ∈ A or to the boundary of the rectangle it is located in. The algorithm involves iteratively merging pairs of rectangles by erasing the shared boundary between them and recomputing the minimum-cost partial matching. This continues until all boundaries are erased and we obtain the desired minimum-cost partial matching of A and B. We exploit geometry in our analysis to show that each point participates in only Õ(n^{1-1/(2d+1)}log Δ) number of augmenting paths, leading to a total execution time of Õ(n^{2-1/(2d+1)}Φ(n)log Δ).
We also show that, for the 𝓁₁ norm and d dimensions, any algorithm that can solve instances of the offline n-server problem with an exponential spread in T(n) time can be used to compute minimum-cost bipartite matching in a complete graph defined on two (d-1)-dimensional point sets under the 𝓁₁ norm within T(n) time. This suggests that removing spread from the execution time of our algorithm may be difficult as it immediately results in a sub-quadratic algorithm for bipartite matching under the 𝓁₁ norm.
Cite as
Sharath Raghvendra, Pouyan Shirzadian, and Rachita Sowle. Geometric Bipartite Matching Based Exact Algorithms for Server Problems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 72:1-72:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{raghvendra_et_al:LIPIcs.SoCG.2025.72,
author = {Raghvendra, Sharath and Shirzadian, Pouyan and Sowle, Rachita},
title = {{Geometric Bipartite Matching Based Exact Algorithms for Server Problems}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {72:1--72:15},
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.2025.72},
URN = {urn:nbn:de:0030-drops-232240},
doi = {10.4230/LIPIcs.SoCG.2025.72},
annote = {Keywords: Minimum-Cost Bipartite Matching, Server Problems, Primal-Dual Approach}
}
Document
Embedding Graphs as Euclidean kNN-Graphs
Abstract
Let G = (V,E) be a directed graph on n vertices where each vertex has out-degree k. We say that G is kNN-realizable in d-dimensional Euclidean space if there exists a point set P = {p_1, p_2, …, p_n} in ℝ^d along with a one-to-one mapping ϕ: V → P such that for any u,v ∈ V, u is an out-neighbor of v in G if and only if ϕ(u) is one of the k nearest neighbors of ϕ(v); we call the map ϕ a kNN-realization of G in ℝ^d. The kNN-realization problem, which aims to compute a kNN-realization of an input graph in ℝ^d, is known to be NP-hard already for d = 2 and k = 1 [Eades and Whitesides, Theoretical Computer Science, 1996], and to the best of our knowledge has not been studied in dimension d = 1. The main results of this paper are the following:
- For any fixed dimension d ≥ 2, we can efficiently compute an embedding realizing at least a 1 - ε fraction of G’s edges, or conclude that G is not kNN-realizable in ℝ^d.
- For d = 1, we can decide in O(kn) time whether G is kNN-realizable and, if so, compute a realization in O(n^{2.5} poly(log n)) time.
Cite as
Thomas Schibler, Subhash Suri, and Jie Xue. Embedding Graphs as Euclidean kNN-Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 73:1-73:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{schibler_et_al:LIPIcs.SoCG.2025.73,
author = {Schibler, Thomas and Suri, Subhash and Xue, Jie},
title = {{Embedding Graphs as Euclidean kNN-Graphs}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {73:1--73:14},
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.2025.73},
URN = {urn:nbn:de:0030-drops-232253},
doi = {10.4230/LIPIcs.SoCG.2025.73},
annote = {Keywords: Geometric graphs, k-nearest neighbors, graph embedding, approximation algorithms}
}
Document
The Point-Boundary Art Gallery Problem Is ∃ℝ-Hard
Abstract
We resolve the complexity of the point-boundary variant of the art gallery problem, showing that it is ∃ℝ-complete, meaning that it is equivalent under polynomial time reductions to deciding whether a system of polynomial equations has a real solution.
The art gallery problem asks whether there is a configuration of guards that together can see every point inside of an art gallery modeled by a simple polygon. The original version of this problem (which we call the point-point variant) was shown to be ∃ℝ-hard [Abrahamsen, Adamaszek, and Miltzow, JACM 2021], but the complexity of the variant where guards only need to guard the walls of the art gallery was left as an open problem. We show that this variant is also ∃ℝ-hard.
Our techniques can also be used to greatly simplify the proof of ∃ℝ-hardness of the point-point art gallery problem. The gadgets in previous work could only be constructed by using a computer to find complicated rational coordinates with specific algebraic properties. All of our gadgets can be constructed by hand and can be verified with simple geometric arguments.
Cite as
Jack Stade. The Point-Boundary Art Gallery Problem Is ∃ℝ-Hard. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 74:1-74:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{stade:LIPIcs.SoCG.2025.74,
author = {Stade, Jack},
title = {{The Point-Boundary Art Gallery Problem Is \exists\mathbb{R}-Hard}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {74:1--74:23},
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.2025.74},
URN = {urn:nbn:de:0030-drops-232269},
doi = {10.4230/LIPIcs.SoCG.2025.74},
annote = {Keywords: Art Gallery Problem, Complexity, ETR, Polygon}
}
Document
Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers
Abstract
A long-standing conjecture of Eckhoff, Linhart, and Welzl, which would generalize McMullen’s Upper Bound Theorem for polytopes and refine asymptotic bounds due to Clarkson, asserts that for k ⩽ ⌊(n-d-2)/2⌋, the complexity of the (⩽ k)-level in a simple arrangement of n hemispheres in S^d is maximized for arrangements that are polar duals of neighborly d-polytopes. We prove this conjecture in the case n = d+4. By Gale duality, this implies the following result about crossing numbers: In every spherical arc drawing of K_n in S² (given by a set V ⊂ S² of n unit vectors connected by spherical arcs), the number of crossings is at least 1/4 ⌊n/2⌋ ⌊(n-1)/2⌋ ⌊(n-2)/2⌋ ⌊(n-3)/2⌋. This lower bound is attained if every open linear halfspace contains at least ⌊(n-2)/2⌋ of the vectors in V.
Moreover, we determine the space of all linear and affine relations that hold between the face numbers of levels in simple arrangements of n hemispheres in S^d. This completes a long line of research on such relations, answers a question posed by Andrzejak and Welzl in 2003, and generalizes the classical fact that the Dehn-Sommerville relations generate all linear relations between the face numbers of simple polytopes (which correspond to the 0-level).
To prove these results, we introduce the notion of the g-matrix, which encodes the face numbers of levels in an arrangement and generalizes the classical g-vector of a polytope.
Cite as
Elizaveta Streltsova and Uli Wagner. Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 75:1-75:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{streltsova_et_al:LIPIcs.SoCG.2025.75,
author = {Streltsova, Elizaveta and Wagner, Uli},
title = {{Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {75:1--75:18},
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.2025.75},
URN = {urn:nbn:de:0030-drops-232276},
doi = {10.4230/LIPIcs.SoCG.2025.75},
annote = {Keywords: Levels in arrangements, k-sets, k-facets, convex polytopes, f-vector, h-vector, g-vector, Dehn-Sommerville relations, Radon partitions, Gale duality, g-matrix}
}
Document
A Note on the No-(d+2)-On-a-Sphere Problem
Abstract
For fixed d ≥ 3, we construct subsets of the d-dimensional lattice cube [n]^d of size n^{3/(d + 1) - o(1)} with no d+2 points on a sphere or a hyperplane. This improves the previously best known bound of Ω(n^{1/(d-1)}) due to Thiele from 1995.
Cite as
Andrew Suk and Ethan Patrick White. A Note on the No-(d+2)-On-a-Sphere Problem. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 76:1-76:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{suk_et_al:LIPIcs.SoCG.2025.76,
author = {Suk, Andrew and White, Ethan Patrick},
title = {{A Note on the No-(d+2)-On-a-Sphere Problem}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {76:1--76:8},
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.2025.76},
URN = {urn:nbn:de:0030-drops-232287},
doi = {10.4230/LIPIcs.SoCG.2025.76},
annote = {Keywords: General position, no-four-on-a-cirle, d-dimensional lattice cube}
}
Document
Dynamic Maximum Depth of Geometric Objects
Abstract
Given a set of geometric objects in the plane (rectangles, squares, disks etc.), its maximum depth (or geometric clique) is the largest number of objects with a common intersection. In this paper, we present data structures for dynamically maintaining the maximum depth under insertions and deletions of geometric objects, with sublinear update time. We achieve the following results:
- a 1/k-approximate dynamic maximum-depth data structure for (axis-parallel) rectangles with O(n^{1/(k+1)} log n) amortized update time, for any fixed k ∈ ℤ^+. In particular, when k = 1, this gives an exact data structure for rectangles with O(√n log n) amortized update time, almost matching the best known bound for the offline version of the problem.
- a (1/2-ε)-approximate dynamic maximum-depth data structure for disks with n^{2/3} log^{O(1)}n amortized update time, for any constant ε > 0. Having exact data structures for disks with sublinear update time is unlikely, since the static maximum-depth problem for disks is 3SUM-hard and thus does not admit subquadratic-time algorithms.
Cite as
Subhash Suri, Jie Xue, Xiongxin Yang, and Jiumu Zhu. Dynamic Maximum Depth of Geometric Objects. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 77:1-77:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{suri_et_al:LIPIcs.SoCG.2025.77,
author = {Suri, Subhash and Xue, Jie and Yang, Xiongxin and Zhu, Jiumu},
title = {{Dynamic Maximum Depth of Geometric Objects}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {77:1--77:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.77},
URN = {urn:nbn:de:0030-drops-232295},
doi = {10.4230/LIPIcs.SoCG.2025.77},
annote = {Keywords: dynamic algorithms, maximum depth}
}
Document
Efficient Greedy Discrete Subtrajectory Clustering
Abstract
We cluster a set of trajectories 𝒯 using subtrajectories of 𝒯. We require for a clustering C that any two subtrajectories (𝒯[a, b], 𝒯[c, d]) in a cluster have disjoint intervals [a,b] and [c, d]. Clustering quality may be measured by the number of clusters, the number of vertices of 𝒯 that are absent from the clustering, and by the Fréchet distance between subtrajectories in a cluster.
A Δ-cluster of 𝒯 is a cluster 𝒫 of subtrajectories of 𝒯 with a centre P ∈ 𝒫, where all subtrajectories in 𝒫 have Fréchet distance at most Δ to P. Buchin, Buchin, Gudmundsson, Löffler and Luo present two O(n² + n m 𝓁)-time algorithms: SC(max, 𝓁, Δ, 𝒯) computes a single Δ-cluster where P has at least 𝓁 vertices and maximises the cardinality m of 𝒫. SC(m, max, Δ, 𝒯) computes a single Δ-cluster where 𝒫 has cardinality m and maximises the complexity 𝓁 of P.
In this paper, which is a mixture of algorithms engineering and theoretical insights, we use such maximum-cardinality clusters in a greedy clustering algorithm. We first provide an efficient implementation of SC(max, 𝓁, Δ, 𝒯) and SC(m, max, Δ, 𝒯) that significantly outperforms previous implementations. Next, we use these functions as a subroutine in a greedy clustering algorithm, which performs well when compared to existing subtrajectory clustering algorithms on real-world data. Finally, we observe that, for fixed Δ and 𝒯, these two functions always output a point on the Pareto front of some bivariate function θ(𝓁, m). We design a new algorithm PSC(Δ, 𝒯) that in O(n² log⁴ n) time computes a 2-approximation of this Pareto front. This yields a broader set of candidate clusters, with comparable quality to the output of the previous functions. We show that using PSC(Δ, 𝒯) as a subroutine improves the clustering quality and performance even further.
Cite as
Ivor van der Hoog, Lara Ost, Eva Rotenberg, and Daniel Rutschmann. Efficient Greedy Discrete Subtrajectory Clustering. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 78:1-78:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{vanderhoog_et_al:LIPIcs.SoCG.2025.78,
author = {van der Hoog, Ivor and Ost, Lara and Rotenberg, Eva and Rutschmann, Daniel},
title = {{Efficient Greedy Discrete Subtrajectory Clustering}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {78:1--78:20},
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.2025.78},
URN = {urn:nbn:de:0030-drops-232308},
doi = {10.4230/LIPIcs.SoCG.2025.78},
annote = {Keywords: Algorithms engineering, Fr\'{e}chet distance, subtrajectory clustering}
}
Document
CG Challenge
Computing Non-Obtuse Triangulations with Few Steiner Points (CG Challenge)
Abstract
We present the winning implementation of the Seventh Computational Geometry Challenge (CG:SHOP 2025). The task in this challenge was to find non-obtuse triangulations for given planar regions, respecting a given set of constraints consisting of extra vertices and edges that must be part of the triangulation. The goal was to minimize the number of introduced Steiner points. Our approach is to maintain a constrained Delaunay triangulation, for which we repeatedly remove, relocate, or add Steiner points. We use local search to choose the action that improves the triangulation the most, until the resulting triangulation is non-obtuse.
Cite as
Mikkel Abrahamsen, Florestan Brunck, Jacobus Conradi, Benedikt Kolbe, and André Nusser. Computing Non-Obtuse Triangulations with Few Steiner Points (CG Challenge). In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 79:1-79:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2025.79,
author = {Abrahamsen, Mikkel and Brunck, Florestan and Conradi, Jacobus and Kolbe, Benedikt and Nusser, Andr\'{e}},
title = {{Computing Non-Obtuse Triangulations with Few Steiner Points}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {79:1--79:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-370-6},
ISSN = {1868-8969},
year = {2025},
volume = {332},
editor = {Aichholzer, Oswin and Wang, Haitao},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.79},
URN = {urn:nbn:de:0030-drops-232311},
doi = {10.4230/LIPIcs.SoCG.2025.79},
annote = {Keywords: non-obtuse triangulation, local search, competition}
}
Document
CG Challenge
Incremental Algorithm and Local Search for Minimum Non-Obtuse Triangulations (CG Challenge)
Abstract
In this year’s CG challenge, the task was to compute a non-obtuse triangulation of given planar regions while minimizing the number of Steiner points. Our team (Gwamegi) used two approaches. The first approach incrementally adds Steiner points on the grid defined by the input points in the planar regions, while maintaining a Delaunay triangulation. The second approach is an iterated local search, which runs insertion and deletion steps alternatingly. In the insertion step, we add a new Steiner point inside a maximal convex subpolygon in the current triangulation. In the deletion step, we remove a number of Steiner points packed in a small region. We use both our approaches to obtain non-obtuse triangulations for all 150 instances. We use our second approach to reduce the number of Steiner points from the non-obtuse triangulations. We have successfully computed non-obtuse triangulations using a sufficiently small number of Steiner points for all instances.
Cite as
Taehoon Ahn, Jaegun Lee, Byeonguk Kang, and Hwi Kim. Incremental Algorithm and Local Search for Minimum Non-Obtuse Triangulations (CG Challenge). In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 80:1-80:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ahn_et_al:LIPIcs.SoCG.2025.80,
author = {Ahn, Taehoon and Lee, Jaegun and Kang, Byeonguk and Kim, Hwi},
title = {{Incremental Algorithm and Local Search for Minimum Non-Obtuse Triangulations}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {80:1--80:8},
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.2025.80},
URN = {urn:nbn:de:0030-drops-232326},
doi = {10.4230/LIPIcs.SoCG.2025.80},
annote = {Keywords: Triangulation, Non-obtuse triangle, Steiner point, Incremental algorithm, Local search}
}
Document
Media Exposition
AmoebotSim 2.0: A Visual Simulation Environment for the Amoebot Model with Reconfigurable Circuits and Joint Movements (Media Exposition)
Abstract
We present AmoebotSim 2.0, a simulation environment for the geometric amoebot model of programmable matter that supports the reconfigurable circuit and joint movement extensions of the model. In the geometric amoebot model, we consider systems of simple computational entities called amoebots in a regular triangular grid environment. We are interested in distributed algorithms that solve coordination and shape formation problems. The reconfigurable circuit and joint movement extensions of the model allow the amoebots to communicate over greater distances and perform more complex movements, overcoming some limitations of the original model. AmoebotSim 2.0 is an open-source simulation environment that supports these extensions and provides a rich graphical interface, flexible simulation features, an extensive API, and comprehensive documentation.
Cite as
Matthias Artmann, Tobias Maurer, Andreas Padalkin, Daniel Warner, and Christian Scheideler. AmoebotSim 2.0: A Visual Simulation Environment for the Amoebot Model with Reconfigurable Circuits and Joint Movements (Media Exposition). In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 81:1-81:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{artmann_et_al:LIPIcs.SoCG.2025.81,
author = {Artmann, Matthias and Maurer, Tobias and Padalkin, Andreas and Warner, Daniel and Scheideler, Christian},
title = {{AmoebotSim 2.0: A Visual Simulation Environment for the Amoebot Model with Reconfigurable Circuits and Joint Movements}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {81:1--81:5},
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.2025.81},
URN = {urn:nbn:de:0030-drops-232338},
doi = {10.4230/LIPIcs.SoCG.2025.81},
annote = {Keywords: Programmable matter, amoebot model, reconfigurable circuits, joint movements, simulator}
}
Document
Media Exposition
Software for the Thompson and Funk Polygonal Geometry (Media Exposition)
Abstract
Metric spaces defined within convex polygons, such as the Thompson, Funk, reverse Funk, and Hilbert metrics, are subjects of recent exploration and study in computational geometry. This paper contributes an educational piece of software for understanding these unique geometries while also providing a tool to support their research. We provide dynamic software for manipulating the Funk, reverse Funk, and Thompson balls in convex polygonal domains. Additionally, we provide a visualization program for traversing the Hilbert polygonal geometry.
Cite as
Hridhaan Banerjee, Carmen Isabel Day, Auguste H. Gezalyan, Olga Golovatskaia, Megan Hunleth, Sarah Hwang, Nithin Parepally, Lucy Wang, and David M. Mount. Software for the Thompson and Funk Polygonal Geometry (Media Exposition). In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 82:1-82:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{banerjee_et_al:LIPIcs.SoCG.2025.82,
author = {Banerjee, Hridhaan and Day, Carmen Isabel and Gezalyan, Auguste H. and Golovatskaia, Olga and Hunleth, Megan and Hwang, Sarah and Parepally, Nithin and Wang, Lucy and Mount, David M.},
title = {{Software for the Thompson and Funk Polygonal Geometry}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {82:1--82:6},
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.2025.82},
URN = {urn:nbn:de:0030-drops-232349},
doi = {10.4230/LIPIcs.SoCG.2025.82},
annote = {Keywords: Thompson metric, Hilbert metric, Funk metric, balls}
}
Document
Media Exposition
French Onion Soup, Ipelets for Points and Polygons (Media Exposition)
Abstract
There are many structures, both classical and modern, involving point-sets and polygons whose deeper understanding can be facilitated through interactive visualizations. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating geometric figures. One of its features is the capability to extend its functionality through programs called Ipelets. In this media submission, we showcase a collection of new Ipelets that construct a variety of geometric structures based on point sets and polygons. These include quadtrees, trapezoidal maps, beta skeletons, floating bodies of convex polygons, onion graphs, fractals (Sierpiński triangle and carpet), simple polygon triangulations, and random point sets in simple polygons. All our Ipelets are programmed in Lua and are freely available.
Cite as
Klint Faber, Auguste H. Gezalyan, Adam Martinson, Aniruddh Mutnuru, Nithin Parepally, Ryan Parker, Mihil Sreenilayam, Aram Zaprosyan, and David M. Mount. French Onion Soup, Ipelets for Points and Polygons (Media Exposition). In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 83:1-83:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{faber_et_al:LIPIcs.SoCG.2025.83,
author = {Faber, Klint and Gezalyan, Auguste H. and Martinson, Adam and Mutnuru, Aniruddh and Parepally, Nithin and Parker, Ryan and Sreenilayam, Mihil and Zaprosyan, Aram and Mount, David M.},
title = {{French Onion Soup, Ipelets for Points and Polygons}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {83:1--83:6},
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.2025.83},
URN = {urn:nbn:de:0030-drops-232350},
doi = {10.4230/LIPIcs.SoCG.2025.83},
annote = {Keywords: Hilbert metric, Macbeath Regions, Polar Bodies, convexity}
}
Document
Media Exposition
Incremental and Interactive PQ- and PC-Trees (Media Exposition)
Abstract
PQ-trees [Booth, 1975; Kellogg S. Booth and George S. Lueker, 1976] (and their more general variant PC-trees [Hsu and McConnell, 2003; Wei-Kuan and Wen-Lian, 1999]) are a well-known data structure for representing the set of linear (or for PC-trees, circular) orders respecting a given set of consecutivity constraints. Each such constraint is specified by a set of elements and requires that these elements appear consecutively in the linear (or circular) order; thus, they disallow the set to be interleaved with its complement. The main operation supported by these data structures is thus the so-called update, which takes as input a set that forms an additional constraint, and in response changes the tree in order to restrict the represented orders to those satisfying the new constraint. Interpreting a given tree is straightforward: leaves represent the underlying elements, while inner nodes either allow the order of their subtrees to be reversed (Q/C-nodes) or to be arbitrarily permuted (P-nodes). However, the way this structure and the set of represented orders change under updates is less intuitive. We present an interactive web app that allows users to specify sets of consecutivity constraints in the form of a 0/1-matrix and then calculates and visualizes the corresponding PQ- or PC-tree. The constraints can then be changed dynamically while observing how this changes the structure of the tree and the set of represented orders. Through this interactive exploration, we hope to make PQ- and PC-trees more accessible to a wider audience.
Cite as
Simon D. Fink and Dominik Peters. Incremental and Interactive PQ- and PC-Trees (Media Exposition). In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 84:1-84:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{fink_et_al:LIPIcs.SoCG.2025.84,
author = {Fink, Simon D. and Peters, Dominik},
title = {{Incremental and Interactive PQ- and PC-Trees}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {84:1--84:4},
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.2025.84},
URN = {urn:nbn:de:0030-drops-232365},
doi = {10.4230/LIPIcs.SoCG.2025.84},
annote = {Keywords: PQ trees, PC trees, planarity, consecutive ones problem, interactive exploration}
}
Document
Media Exposition
Finding Shortest Reconfiguration Sequences for Modular Robots (Media Exposition)
Abstract
This paper introduces a set of tools built to help researchers design algorithms for modular robots. These tools can brute force solutions to specific reconfigurations, visualize movements of modular robots, and can be used to design specific configurations of robots. Multiple models of modular robots are supported, and can be added by users.
Cite as
UML Modular Robotics Group, Hugo A. Akitaya, Andrew Clements, Sam Downey, Jonathan Eisenbies, Soham Samanta, Gabriel Shahrouzi, and Frederick Stock. Finding Shortest Reconfiguration Sequences for Modular Robots (Media Exposition). In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 85:1-85:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{umlmodularroboticsgroup_et_al:LIPIcs.SoCG.2025.85,
author = {UML Modular Robotics Group and A. Akitaya, Hugo and Clements, Andrew and Downey, Sam and Eisenbies, Jonathan and Samanta, Soham and Shahrouzi, Gabriel and Stock, Frederick},
title = {{Finding Shortest Reconfiguration Sequences for Modular Robots}},
booktitle = {41st International Symposium on Computational Geometry (SoCG 2025)},
pages = {85:1--85:5},
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.2025.85},
URN = {urn:nbn:de:0030-drops-232371},
doi = {10.4230/LIPIcs.SoCG.2025.85},
annote = {Keywords: modular reconfigurable robots, sliding cube model, reconfiguration}
}