22 Search Results for "Walczak, Bartosz"

Document
DOI: 10.4230/LIPIcs.SWAT.2026.6
How Many Slopes Does Polynomial Area Cost?

Authors: Michael A. Bekos, Eleni Katsanou, Philipp Kindermann, and Maria Eleni Pavlidi

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract
In this work, we study the interplay between the number of slopes, the number of bends per edge, and the area requirements for planar drawings of bounded-degree graphs. Our motivation stems from the fact that, while numerous algorithms produce planar drawings with few slopes for graphs of relatively small degree in polynomial area, existing approaches for higher-degree graphs often require super-polynomial area. We address this gap in the literature by presenting new constructions that yield polynomial-area drawings with few bends per edge while slightly increasing the required number of slopes, thereby providing the first systematic study of slopes, bends and area trade-offs.
Cite as

Michael A. Bekos, Eleni Katsanou, Philipp Kindermann, and Maria Eleni Pavlidi. How Many Slopes Does Polynomial Area Cost?. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bekos_et_al:LIPIcs.SWAT.2026.6,
  author =	{Bekos, Michael A. and Katsanou, Eleni and Kindermann, Philipp and Pavlidi, Maria Eleni},
  title =	{{How Many Slopes Does Polynomial Area Cost?}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.6},
  URN =		{urn:nbn:de:0030-drops-260424},
  doi =		{10.4230/LIPIcs.SWAT.2026.6},
  annote =	{Keywords: k-bend planar drawings, planar slope number, area requirements}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2026.40
Maximum Independent Sets in Disk Graphs with Disks in Convex Position

Authors: Anastasiia Tkachenko and Haitao Wang

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract
For a set 𝒟 of disks in the plane, its disk graph G(𝒟) is the graph with vertex set 𝒟, where two vertices are adjacent if and only if the corresponding disks intersect. Given a set 𝒟 of n weighted disks, computing a maximum independent set of G(𝒟) is NP-hard. In this paper, we present an O(n³log n)-time algorithm for this problem in a special setting in which the disks are in convex position, meaning that every disk appears on the convex hull of 𝒟. This setting has been studied previously for disks of equal radius, for which an O(n^{37/11})-time algorithm was known. Our algorithm also works in the weighted case where disks have weights and the goal is to compute a maximum-weight independent set. As an application of our result, we obtain an O(n³log² n)-time algorithm for the dispersion problem on a set of n disks in convex position: given an integer k, compute a subset of k disks that maximizes the minimum pairwise distance among all disks in the subset.
Cite as

Anastasiia Tkachenko and Haitao Wang. Maximum Independent Sets in Disk Graphs with Disks in Convex Position. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{tkachenko_et_al:LIPIcs.SWAT.2026.40,
  author =	{Tkachenko, Anastasiia and Wang, Haitao},
  title =	{{Maximum Independent Sets in Disk Graphs with Disks in Convex Position}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{40:1--40:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.40},
  URN =		{urn:nbn:de:0030-drops-260766},
  doi =		{10.4230/LIPIcs.SWAT.2026.40},
  annote =	{Keywords: disk graphs, independent sets, convex position, dispersion}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2025.17
Treedepth Inapproximability and Exponential ETH Lower Bound

Authors: Édouard Bonnet, Daniel Neuen, and Marek Sokołowski

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a 2^{O(k²)} n-time exact algorithm and a polynomial-time O(OPT log^{3/2} OPT)-approximation algorithm, where the former algorithm returns an elimination forest of height k (witnessing that treedepth is at most k) for the n-vertex input graph G, or correctly reports that G has treedepth larger than k, and OPT is the actual value of the treedepth. On the complexity side, exactly computing treedepth is NP-complete, but the known reductions do not rule out a polynomial-time approximation scheme (PTAS), and under the Exponential Time Hypothesis (ETH) only exclude a running time of 2^o(√n) for exact algorithms. We show that 1.0003-approximating Treedepth is NP-hard, and that exactly computing the treedepth of an n-vertex graph requires time 2^Ω(n), unless the ETH fails. We further derive that there exist absolute constants δ, c > 0 such that any (1+δ)-approximation algorithm requires time 2^Ω(n/log^c n). We do so via a simple direct reduction from Satisfiability to Treedepth, inspired by a reduction recently designed for Treewidth [STOC '25].
Cite as

Édouard Bonnet, Daniel Neuen, and Marek Sokołowski. Treedepth Inapproximability and Exponential ETH Lower Bound. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 17:1-17:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bonnet_et_al:LIPIcs.IPEC.2025.17,
  author =	{Bonnet, \'{E}douard and Neuen, Daniel and Soko{\l}owski, Marek},
  title =	{{Treedepth Inapproximability and Exponential ETH Lower Bound}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{17:1--17:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.17},
  URN =		{urn:nbn:de:0030-drops-251494},
  doi =		{10.4230/LIPIcs.IPEC.2025.17},
  annote =	{Keywords: treedepth, lower bounds, approximation}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.20
Sparse Induced Subgraphs in P₇-Free Graphs of Bounded Clique Number

Authors: Maria Chudnovsky, Jadwiga Czyżewska, Kacper Kluk, Marcin Pilipczuk, and Paweł Rzążewski

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

Abstract
Many natural computational problems, including e.g. Max Weight Independent Set, Feedback Vertex Set, or Vertex Planarization, can be unified under an umbrella of finding the largest sparse induced subgraph that satisfies some property definable in CMSO₂ logic. It is believed that each problem expressible with this formalism can be solved in polynomial time in graphs that exclude a fixed path as an induced subgraph. This belief is supported by the existence of a quasipolynomial-time algorithm by Gartland, Lokshtanov, Pilipczuk, Pilipczuk, and Rzążewski [STOC 2021], and a recent polynomial-time algorithm for P₆-free graphs by Chudnovsky, McCarty, Pilipczuk, Pilipczuk, and Rzążewski [SODA 2024]. In this work we extend polynomial-time tractability of all such problems to P₇-free graphs of bounded clique number.
Cite as

Maria Chudnovsky, Jadwiga Czyżewska, Kacper Kluk, Marcin Pilipczuk, and Paweł Rzążewski. Sparse Induced Subgraphs in P₇-Free Graphs of Bounded Clique Number. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chudnovsky_et_al:LIPIcs.ISAAC.2025.20,
  author =	{Chudnovsky, Maria and Czy\.{z}ewska, Jadwiga and Kluk, Kacper and Pilipczuk, Marcin and Rz\k{a}\.{z}ewski, Pawe{\l}},
  title =	{{Sparse Induced Subgraphs in P₇-Free Graphs of Bounded Clique Number}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{20:1--20:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.20},
  URN =		{urn:nbn:de:0030-drops-249282},
  doi =		{10.4230/LIPIcs.ISAAC.2025.20},
  annote =	{Keywords: P\underlinet-free graphs, maximum weight induced subgraph, maximum weight independent set}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.25
Structural Parameterizations of Simultaneous Planarity

Authors: Thomas Depian, Simon D. Fink, Alexander Firbas, Robert Ganian, Matthias Pfretzschner, and Ignaz Rutter

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

Abstract
Given a set of graphs on the same vertex set, the problem Simultaneous Embedding With Fixed Edges (SEFE) asks, whether there exist planar drawings of all input graphs, such that every pair of drawings coincides on their shared subgraph. It is known that SEFE is NP-complete [Elisabeth Gassner et al., 2006], even in the so-called sunflower case, where all pairs of input graphs have the same shared graph G_∩ [Marcus Schaefer, 2012]. Fink, Pfretzschner, and Rutter [Simon D. Fink et al., 2023] recently initiated the study of the parameterized complexity of SEFE in the sunflower case, mainly focusing on structural parameters of G_∩. In this work, we shift the focus towards parameters of the union graph G_∪ that contains the edges of all input graphs. On the positive side, we establish fixed-parameter tractability for the problem with respect to the feedback edge set number of G_∪. We complement this result by showing that it, surprisingly, remains NP-complete even if G_∪ has constant vertex cover number. These results settle two open questions posed by Fink et al. [Simon D. Fink et al., 2023].
Cite as

Thomas Depian, Simon D. Fink, Alexander Firbas, Robert Ganian, Matthias Pfretzschner, and Ignaz Rutter. Structural Parameterizations of Simultaneous Planarity. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{depian_et_al:LIPIcs.ISAAC.2025.25,
  author =	{Depian, Thomas and Fink, Simon D. and Firbas, Alexander and Ganian, Robert and Pfretzschner, Matthias and Rutter, Ignaz},
  title =	{{Structural Parameterizations of Simultaneous Planarity}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.25},
  URN =		{urn:nbn:de:0030-drops-249332},
  doi =		{10.4230/LIPIcs.ISAAC.2025.25},
  annote =	{Keywords: SEFE, Simultaneous Planarity, Fixed-Parameter Tractability, NP-hardness}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.67
Compact Representation of Semilinear and Terrain-Like Graphs

Authors: Jean Cardinal and Yelena Yuditsky

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

Abstract
We consider the existence and construction of biclique covers of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The size of such a cover is the sum of the sizes of the bicliques. Small-size biclique covers of graphs are ubiquitous in computational geometry, and have been shown to be useful compact representations of graphs. We give a brief survey of classical and recent results on biclique covers and their applications, and give new families of graphs having biclique covers of near-linear size. In particular, we show that semilinear graphs, whose edges are defined by linear relations in bounded dimensional space, always have biclique covers of size O(npolylog n). This generalizes many previously known results on special classes of graphs including interval graphs, permutation graphs, and graphs of bounded boxicity, but also new classes such as intersection graphs of L-shapes in the plane. It also directly implies the bounds for Zarankiewicz’s problem derived by Basit, Chernikov, Starchenko, Tao, and Tran (Forum Math. Sigma, 2021). We also consider capped graphs, also known as terrain-like graphs, defined as ordered graphs forbidding a certain ordered pattern on four vertices. Terrain-like graphs contain the induced subgraphs of terrain visibility graphs. We give an elementary proof that these graphs admit biclique partitions of size O(nlog³ n). This provides a simple combinatorial analogue of a classical result from Agarwal, Alon, Aronov, and Suri on polygon visibility graphs (Discrete Comput. Geom. 1994). Finally, we prove that there exists families of unit disk graphs on n vertices that do not admit biclique coverings of size o(n^{4/3}), showing that we are unlikely to improve on Szemerédi-Trotter type incidence bounds for higher-degree semialgebraic graphs.
Cite as

Jean Cardinal and Yelena Yuditsky. Compact Representation of Semilinear and Terrain-Like Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cardinal_et_al:LIPIcs.ESA.2025.67,
  author =	{Cardinal, Jean and Yuditsky, Yelena},
  title =	{{Compact Representation of Semilinear and Terrain-Like Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{67:1--67:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.67},
  URN =		{urn:nbn:de:0030-drops-245359},
  doi =		{10.4230/LIPIcs.ESA.2025.67},
  annote =	{Keywords: Biclique covers, intersection graphs, visibility graphs, Zarankiewicz’s problem}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.15
A Polynomial-Time Approximation Algorithm for Complete Interval Minors

Authors: Romain Bourneuf, Julien Cocquet, Chaoliang Tang, and Stéphan Thomassé

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
As shown by Robertson and Seymour, deciding whether the complete graph K_t is a minor of an input graph G is a fixed parameter tractable problem when parameterized by t. From the approximation viewpoint, a substantial gap remains: there is no PTAS for finding the largest complete minor unless P = NP, whereas the best known result is a polytime O(√ n)-approximation algorithm by Alon, Lingas and Wahlén. We investigate the complexity of finding K_t as interval minor in ordered graphs (i.e. graphs with a linear order on the vertices, in which intervals are contracted to form minors). Our main result is a polytime f(t)-approximation algorithm, where f is triply exponential in t but independent of n. The algorithm is based on delayed decompositions and shows that ordered graphs without a K_t interval minor can be constructed via a bounded number of three operations: closure under substitutions, edge union, and concatenation of a stable set. As a byproduct, graphs avoiding K_t as an interval minor have bounded chromatic number.
Cite as

Romain Bourneuf, Julien Cocquet, Chaoliang Tang, and Stéphan Thomassé. A Polynomial-Time Approximation Algorithm for Complete Interval Minors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 15:1-15:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bourneuf_et_al:LIPIcs.APPROX/RANDOM.2025.15,
  author =	{Bourneuf, Romain and Cocquet, Julien and Tang, Chaoliang and Thomass\'{e}, St\'{e}phan},
  title =	{{A Polynomial-Time Approximation Algorithm for Complete Interval Minors}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{15:1--15:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.15},
  URN =		{urn:nbn:de:0030-drops-243814},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.15},
  annote =	{Keywords: Approximation algorithm, Ordered graphs, Interval minors, Delayed decompositions}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.16
An Improved Guillotine Cut for Squares

Authors: Parinya Chalermsook, Axel Kugelmann, Ly Orgo, Sumedha Uniyal, and Minoo Zarsav

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

Abstract
Given a set of n non-overlapping geometric objects, can we separate a constant fraction of them using straight-line cuts that extend from edge to edge? In 1996, Urrutia posed this question for compact convex objects. Pach and Tardos later refuted it for general line segments by constructing a family where any separable subfamily has size at most O (n^{log₃ 2}). However, for axis-parallel rectangles, they provided positive evidence, showing that an Ω(1/log n)-fraction can be separated. This problem naturally arises in geometric approximation algorithms. In particular, when restricting cuts to only orthogonal straight lines, known as a guillotine cut sequence, any bound on the separability ratio directly translates into a clean and simple dynamic programming for computing a maximum independent set of geometric objects. This paper focuses on the case when the objects are squares. For squares of arbitrary sizes, an Ω(1)-fraction can be separated (Abed et al., APPROX 2015), recently improved to 1/40 (and 1/160 ≈ 0.62% for the weighted case) (Khan and Pittu, APPROX 2020). We further improve this bound, showing that a 9/256 ≈ 3.51% can be separated for the weighted case. This result significantly narrows the possible range for squares to [3.51%, 50%]. The key to our improvement is a refined analysis of the existing framework.
Cite as

Parinya Chalermsook, Axel Kugelmann, Ly Orgo, Sumedha Uniyal, and Minoo Zarsav. An Improved Guillotine Cut for Squares. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chalermsook_et_al:LIPIcs.WADS.2025.16,
  author =	{Chalermsook, Parinya and Kugelmann, Axel and Orgo, Ly and Uniyal, Sumedha and Zarsav, Minoo},
  title =	{{An Improved Guillotine Cut for Squares}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{16:1--16:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.16},
  URN =		{urn:nbn:de:0030-drops-242472},
  doi =		{10.4230/LIPIcs.WADS.2025.16},
  annote =	{Keywords: Guillotine cuts, Geometric Approximation Algorithms, Rectangles, Squares}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.36
Strongly Sublinear Separators and Bounded Asymptotic Dimension for Sphere Intersection Graphs

Authors: James Davies, Agelos Georgakopoulos, Meike Hatzel, and Rose McCarty

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

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
DOI: 10.4230/LIPIcs.SoCG.2025.20
Polynomial-Time Algorithms for Contiguous Art Gallery and Related Problems

Authors: 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

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

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
DOI: 10.4230/LIPIcs.SoCG.2025.24
An 11/6-Approximation Algorithm for Vertex Cover on String Graphs

Authors: Édouard Bonnet and Paweł Rzążewski

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

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
DOI: 10.4230/LIPIcs.SoCG.2024.67
Separator Theorem and Algorithms for Planar Hyperbolic Graphs

Authors: Sándor Kisfaludi-Bak, Jana Masaříková, Erik Jan van Leeuwen, Bartosz Walczak, and Karol Węgrzycki

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

Abstract
The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity. Our main technical contribution is a novel balanced separator theorem for planar δ-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed δ ⩾ 0, we can find a small balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph. An important advantage of our separator is that the union of our separator (vertex set Z) with any subset of the connected components of G - Z induces again a planar δ-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that the size of the separator is poly(δ) ⋅ log n. As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar δ-hyperbolic graphs. We prove that both Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant δ, running in n polylog(n) ⋅ 2^𝒪(δ²) ⋅ ε^{-𝒪(δ)} time. We also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no n^{o(δ)}-time algorithm on planar δ-hyperbolic graphs, unless ETH fails.
Cite as

Sándor Kisfaludi-Bak, Jana Masaříková, Erik Jan van Leeuwen, Bartosz Walczak, and Karol Węgrzycki. Separator Theorem and Algorithms for Planar Hyperbolic Graphs. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 67:1-67:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kisfaludibak_et_al:LIPIcs.SoCG.2024.67,
  author =	{Kisfaludi-Bak, S\'{a}ndor and Masa\v{r}{\'\i}kov\'{a}, Jana and van Leeuwen, Erik Jan and Walczak, Bartosz and W\k{e}grzycki, Karol},
  title =	{{Separator Theorem and Algorithms for Planar Hyperbolic Graphs}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{67:1--67:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.67},
  URN =		{urn:nbn:de:0030-drops-200126},
  doi =		{10.4230/LIPIcs.SoCG.2024.67},
  annote =	{Keywords: Hyperbolic metric, Planar Graphs, r-Division, Approximation Algorithms}
}
Document
Vision
DOI: 10.4230/TGDK.1.1.6
Towards Ordinal Data Science

Authors: Gerd Stumme, Dominik Dürrschnabel, and Tom Hanika

Published in: TGDK, Volume 1, Issue 1 (2023): Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge, Volume 1, Issue 1

Abstract
Order is one of the main instruments to measure the relationship between objects in (empirical) data. However, compared to methods that use numerical properties of objects, the amount of ordinal methods developed is rather small. One reason for this is the limited availability of computational resources in the last century that would have been required for ordinal computations. Another reason - particularly important for this line of research - is that order-based methods are often seen as too mathematically rigorous for applying them to real-world data. In this paper, we will therefore discuss different means for measuring and ‘calculating’ with ordinal structures - a specific class of directed graphs - and show how to infer knowledge from them. Our aim is to establish Ordinal Data Science as a fundamentally new research agenda. Besides cross-fertilization with other cornerstone machine learning and knowledge representation methods, a broad range of disciplines will benefit from this endeavor, including, psychology, sociology, economics, web science, knowledge engineering, scientometrics.
Cite as

Gerd Stumme, Dominik Dürrschnabel, and Tom Hanika. Towards Ordinal Data Science. In Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge (TGDK), Volume 1, Issue 1, pp. 6:1-6:39, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{stumme_et_al:TGDK.1.1.6,
  author =	{Stumme, Gerd and D\"{u}rrschnabel, Dominik and Hanika, Tom},
  title =	{{Towards Ordinal Data Science}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{6:1--6:39},
  ISSN =	{2942-7517},
  year =	{2023},
  volume =	{1},
  number =	{1},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.1.1.6},
  URN =		{urn:nbn:de:0030-drops-194801},
  doi =		{10.4230/TGDK.1.1.6},
  annote =	{Keywords: Order relation, data science, relational theory of measurement, metric learning, general algebra, lattices, factorization, approximations and heuristics, factor analysis, visualization, browsing, explainability}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.2
Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks

Authors: Mikkel Abrahamsen and Bartosz Walczak

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

Abstract
For smooth convex disks A, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes G^{hom}(A) and G^{sim}(A) of intersection graphs that can be obtained from homothets and similarities of A, respectively. Namely, we prove that G^{hom}(A) = G^{hom}(B) if and only if A and B are affine equivalent, and G^{sim}(A) = G^{sim}(B) if and only if A and B are similar.
Cite as

Mikkel Abrahamsen and Bartosz Walczak. Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2023.2,
  author =	{Abrahamsen, Mikkel and Walczak, Bartosz},
  title =	{{Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{2:1--2:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.2},
  URN =		{urn:nbn:de:0030-drops-178523},
  doi =		{10.4230/LIPIcs.SoCG.2023.2},
  annote =	{Keywords: geometric intersection graph, convex disk, homothet, similarity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.33
A Solution to Ringel’s Circle Problem

Authors: James Davies, Chaya Keller, Linda Kleist, Shakhar Smorodinsky, and Bartosz Walczak

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

Abstract
We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel’s circle problem (1959). The proof relies on a (multidimensional) version of Gallai’s theorem with polynomial constraints, which we derive from the Hales-Jewett theorem and which may be of independent interest.
Cite as

James Davies, Chaya Keller, Linda Kleist, Shakhar Smorodinsky, and Bartosz Walczak. A Solution to Ringel’s Circle Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{davies_et_al:LIPIcs.SoCG.2022.33,
  author =	{Davies, James and Keller, Chaya and Kleist, Linda and Smorodinsky, Shakhar and Walczak, Bartosz},
  title =	{{A Solution to Ringel’s Circle Problem}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{33:1--33:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.33},
  URN =		{urn:nbn:de:0030-drops-160413},
  doi =		{10.4230/LIPIcs.SoCG.2022.33},
  annote =	{Keywords: circle arrangement, chromatic number, Gallai’s theorem, polynomial method}
}
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