40 Search Results for "Tardos, Gábor"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.2
On the Maximum Number of Tangencies Among 1-Intersecting Curves

Authors: Eyal Ackerman and Balázs Keszegh

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

Abstract
According to a conjecture of Pach, there are O(n) tangent pairs among any family of n Jordan arcs in which every pair of arcs has precisely one common point and no three arcs share a common point. This conjecture was proved for two special cases, however, for the general case the currently best upper bound is only O(n^{7/4}). This is also the best known bound on the number of tangencies in the relaxed case where every pair of arcs has at most one common point. We improve the bounds for the latter and former cases to O(n^{5/3}) and O(n^{3/2}), respectively. We also consider a few other variants of these questions, for example, we show that if the arcs are x-monotone, each pair intersects at most once and their left endpoints lie on a common vertical line, then the maximum number of tangencies is Θ(n^{4/3}). Without this last condition the number of tangencies is O(n^{4/3}(log n)^{1/3}), improving a previous bound of Pach and Sharir. Along the way we prove a graph-theoretic theorem which extends a result of Erdős and Simonovits and may be of independent interest.
Cite as

Eyal Ackerman and Balázs Keszegh. On the Maximum Number of Tangencies Among 1-Intersecting Curves. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ackerman_et_al:LIPIcs.SoCG.2026.2,
  author =	{Ackerman, Eyal and Keszegh, Bal\'{a}zs},
  title =	{{On the Maximum Number of Tangencies Among 1-Intersecting Curves}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{2:1--2:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.2},
  URN =		{urn:nbn:de:0030-drops-258085},
  doi =		{10.4230/LIPIcs.SoCG.2026.2},
  annote =	{Keywords: tangency graph, forbidden subgraph, extremal graph}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.63
Unavoidable Patterns and Plane Paths in Dense Topological Graphs

Authors: Balázs Keszegh, Andrew Suk, Gábor Tardos, and Ji Zeng

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

Abstract
Let C_{s,t} be the complete bipartite geometric graph, with s and t vertices on two distinct parallel lines respectively, and all s t straight-line edges drawn between them. In this paper, we show that every complete bipartite simple topological graph, with parts of size 2(k-1)⁴ + 1 and 2^{k^{5k}}, contains a topological subgraph weakly isomorphic to C_{k,k}. As a corollary, every n-vertex simple topological graph not containing a plane path of length k has at most O_k(n^{2 - 8/k⁴}) edges. When k = 3, we obtain a stronger bound by showing that every n-vertex simple topological graph not containing a plane path of length 3 has at most O(n^{4/3}) edges. We also prove that x-monotone simple topological graphs not containing a plane path of length 3 have at most a linear number of edges.
Cite as

Balázs Keszegh, Andrew Suk, Gábor Tardos, and Ji Zeng. Unavoidable Patterns and Plane Paths in Dense Topological Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{keszegh_et_al:LIPIcs.SoCG.2026.63,
  author =	{Keszegh, Bal\'{a}zs and Suk, Andrew and Tardos, G\'{a}bor and Zeng, Ji},
  title =	{{Unavoidable Patterns and Plane Paths in Dense Topological Graphs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{63:1--63:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.63},
  URN =		{urn:nbn:de:0030-drops-258706},
  doi =		{10.4230/LIPIcs.SoCG.2026.63},
  annote =	{Keywords: graph drawing, topological graph, bipartite geometric graph, forbidden subgraph, extremal graph, thrackle}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.83
Erdős’s Unit Distance Problem and Rigidity

Authors: János Pach, Orit E. Raz, and József Solymosi

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

Abstract
According to a classical result of Spencer, Szemerédi, and Trotter (1984), the maximum number of times the unit distance can occur among n points in the plane is O(n^{4/3}). This is far from Erdős’s lower bound, n^{1+O(1/log log n)}, which is conjectured to be optimal. We prove a structural result for point sets with nearly n^{4/3} unit distances and use it to reduce the problem to a conjecture on rigid frameworks. This conjecture, if true, would yield the first improvement on the bound of Spencer et al. A weaker version of this conjecture has been established by Raz and Solymosi.
Cite as

János Pach, Orit E. Raz, and József Solymosi. Erdős’s Unit Distance Problem and Rigidity. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 83:1-83:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2026.83,
  author =	{Pach, J\'{a}nos and Raz, Orit E. and Solymosi, J\'{o}zsef},
  title =	{{Erd\H{o}s’s Unit Distance Problem and Rigidity}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{83:1--83:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.83},
  URN =		{urn:nbn:de:0030-drops-258906},
  doi =		{10.4230/LIPIcs.SoCG.2026.83},
  annote =	{Keywords: Unit distance problem, Erd\H{o}s, graph rigidity, incidences, polynomial partitioning technique}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.12
Dynamic and Streaming Algorithms for Union Volume Estimation

Authors: Sujoy Bhore, Karl Bringmann, Timothy M. Chan, and Yanheng Wang

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

Abstract
The union volume estimation problem asks to (1±ε)-approximate the volume of the union of n given objects X₁,…,X_n ⊂ ℝ^d. In their seminal work in 1989, Karp, Luby, and Madras solved this problem in time O(n/ε²) in an oracle model where each object X_i can be accessed via three types of queries: obtain the volume of X_i, sample a random point from X_i, and test whether X_i contains a given point x. This running time was recently shown to be optimal [Bringmann, Larsen, Nusser, Rotenberg, and Wang, SoCG'25]. In another line of work, Meel, Vinodchandran, and Chakraborty [PODS'21] designed algorithms that read the objects in one pass using polylogarithmic time per object and polylogarithmic space; this can be phrased as a dynamic algorithm supporting insertions of objects for union volume estimation in the oracle model. In this paper, we study algorithms for union volume estimation in the oracle model that support both insertions and deletions of objects. We obtain the following results: 1) an algorithm supporting insertions and deletions in polylogarithmic update and query time and linear space (this is the first such dynamic algorithm, even for 2D triangles); 2) an algorithm supporting insertions and suffix queries (which generalizes the sliding window setting) in polylogarithmic update and query time and space; 3) an algorithm supporting insertions and deletions of convex bodies of constant dimension in polylogarithmic update and query time and space.
Cite as

Sujoy Bhore, Karl Bringmann, Timothy M. Chan, and Yanheng Wang. Dynamic and Streaming Algorithms for Union Volume Estimation. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.12,
  author =	{Bhore, Sujoy and Bringmann, Karl and Chan, Timothy M. and Wang, Yanheng},
  title =	{{Dynamic and Streaming Algorithms for Union Volume Estimation}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.12},
  URN =		{urn:nbn:de:0030-drops-258180},
  doi =		{10.4230/LIPIcs.SoCG.2026.12},
  annote =	{Keywords: union volume estimation, dynamic algorithms, streaming algorithms}
}
Document
DOI: 10.4230/LIPIcs.FUN.2026.19
Sinks and Ladders: ARRIVAL and SSG with Two Vertices per Level

Authors: Bernd Gärtner, Sebastian Haslebacher, and Hung P. Hoang

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

Abstract
ARRIVAL is the problem of deciding whether a token, following a deterministic process, eventually reaches a designated destination. While the problem is known to lie in NP ∩ CoNP, whether it can be solved in polynomial time remains a major open question. In this article, we study ladders, a class of graphs that constitutes a family of worst-case instances for many existing algorithms, including the currently best known algorithm by Gärtner, Haslebacher, and Hoang (ICALP 2021). We show that ARRIVAL restricted to ladders can be solved in polynomial time, and we further extend this result to stopping binary simple stochastic games (SSG).
Cite as

Bernd Gärtner, Sebastian Haslebacher, and Hung P. Hoang. Sinks and Ladders: ARRIVAL and SSG with Two Vertices per Level. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{gartner_et_al:LIPIcs.FUN.2026.19,
  author =	{G\"{a}rtner, Bernd and Haslebacher, Sebastian and Hoang, Hung P.},
  title =	{{Sinks and Ladders: ARRIVAL and SSG with Two Vertices per Level}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.19},
  URN =		{urn:nbn:de:0030-drops-257385},
  doi =		{10.4230/LIPIcs.FUN.2026.19},
  annote =	{Keywords: ARRIVAL, Rotor-Routing, Simple Stochastic Games}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.55
Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs

Authors: Klim Efremenko, Gillat Kol, Raghuvansh R. Saxena, and Zhijun Zhang

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Sampling a random walk is a fundamental primitive in many graph applications. In the streaming model, it is known that sampling an L-step random walk on an n-vertex directed graph requires Ω(n L) space, implying that no sublinear-space streaming algorithm exists for general graphs. We show that sublinear algorithms are possible for the case of dense graphs, where every vertex has out-degree at least Ω(n). In particular, we give a one-pass turnstile streaming algorithm that uses only 𝒪̃(L) memory for such graphs. More broadly, for graphs with minimum out-degree at least d, our streaming algorithm samples a random walk using 𝒪̃(n/d ⋅ L) memory. We show that our algorithm is optimal in a strong "beyond worst-case" sense. To formalize this, we introduce the notion of universal optimality for graph streaming algorithms. Informally, a streaming algorithm is universally optimal if it performs (almost) as well as possible on every graph, assuming a worst-case choice of the streaming order. This notion of universal optimality is a key conceptual contribution of our work.
Cite as

Klim Efremenko, Gillat Kol, Raghuvansh R. Saxena, and Zhijun Zhang. Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 55:1-55:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{efremenko_et_al:LIPIcs.ITCS.2026.55,
  author =	{Efremenko, Klim and Kol, Gillat and Saxena, Raghuvansh R. and Zhang, Zhijun},
  title =	{{Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{55:1--55:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.55},
  URN =		{urn:nbn:de:0030-drops-253423},
  doi =		{10.4230/LIPIcs.ITCS.2026.55},
  annote =	{Keywords: Random Walk, streaming Algorithm, universal Optimality}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.63
Perfect Simulation of Las Vegas Algorithms via Local Computation

Authors: Xinyu Fu, Yonggang Jiang, and Yitong Yin

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The notion of Las Vegas algorithms was introduced by Babai (1979) and can be defined in two ways: - In Babai’s original definition, a randomized algorithm is called Las Vegas if it has a finitely bounded running time and certifiable random failure. - Another definition widely accepted today is that Las Vegas algorithms refer to zero-error randomized algorithms with random running times. The equivalence between the two definitions is straightforward. Specifically, for randomized algorithms with certifiable failures, repeatedly running the algorithm until no failure is encountered allows for faithful simulation of the correct output when it executes successfully. We show that a similar perfect simulation can also be achieved in distributed local computation. Specifically, in the LOCAL model, with a polylogarithmic overhead in time complexity, any Las Vegas algorithm with finitely bounded running time and locally certifiable failures can be converted to a zero error Las Vegas algorithm. This transformed algorithm faithfully reproduces the correct output of the original algorithm in successful executions. This is achieved by a reduction to a distributed sampling problem under the Lovász Local Lemma (LLL), where the objective is to sample from the joint distribution of random variables avoiding all bad events. We then design the first efficient algorithm to solve this sampling problem in the LOCAL model.
Cite as

Xinyu Fu, Yonggang Jiang, and Yitong Yin. Perfect Simulation of Las Vegas Algorithms via Local Computation. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 63:1-63:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fu_et_al:LIPIcs.ITCS.2026.63,
  author =	{Fu, Xinyu and Jiang, Yonggang and Yin, Yitong},
  title =	{{Perfect Simulation of Las Vegas Algorithms via Local Computation}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{63:1--63:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.63},
  URN =		{urn:nbn:de:0030-drops-253503},
  doi =		{10.4230/LIPIcs.ITCS.2026.63},
  annote =	{Keywords: Las Vegas algorithms, perfect simulation, Lov\'{a}sz Local Lemma, sampling}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2025.21
Overlay Network Construction: Improved Overall and Node-Wise Message Complexity

Authors: Yi-Jun Chang, Yanyu Chen, and Gopinath Mishra

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract
We consider the problem of constructing distributed overlay networks, where nodes in a reconfigurable system can create or sever connections with nodes whose identifiers they know. Initially, each node knows only its own and its neighbors' identifiers, forming a local channel, while the evolving structure is termed the global channel. The goal is to reconfigure any connected graph into a desired topology, such as a bounded-degree expander graph or a well-formed tree (WFT) with a constant maximum degree and logarithmic diameter, minimizing the total number of rounds and message complexity. This problem mirrors real-world peer-to-peer network construction, where creating robust and efficient systems is desired. We study the overlay reconstruction problem in a network of n nodes in two models: GOSSIP-reply and HYBRID. In the GOSSIP-reply model, each node can send a message and receive a corresponding reply message in one round. In the HYBRID model, a node can send O(1) messages to each neighbor in the local channel and a total of O(log n) messages in the global channel. In both models, we propose protocols for WFT construction with O (n log n) message complexities using messages of O(log n) bits. In the GOSSIP-reply model, our protocol takes O(log n) rounds while in the HYBRID model, our protocol takes O(log² n) rounds. Both protocols use O (n log² n) bits of communication. We obtain improved bounds over prior work: GOSSIP-reply: A recent result by Dufoulon et al. (ITCS 2024) achieved O(log⁵ n) round complexity and O (n log⁵ n) message complexity using messages of at least Ω(log² n) bits in GOSSIP-reply. With messages of size O(log n), our protocol achieves an optimal round complexity of O(log n) and an improved message complexity of O(n log n). HYBRID: Götte et al. (Distributed Computing 2023) showed an optimal O(log n)-round algorithm with O(log² n) global messages per round which incurs a message complexity of Ω(m), where m is the number of edges in the initial topology. At the cost of increasing the round complexity to O(log² n) while using only O(log n) messages globally, our protocol achieves a message complexity that is independent of m. Our approach ensures that the total number of messages for node v, with degree deg(v) in the initial topology, is bounded by O(deg(v) + log n), while the algorithm of Götte et al. requires O(deg(v) + (log⁴ n)/(log log n)) messages per node.
Cite as

Yi-Jun Chang, Yanyu Chen, and Gopinath Mishra. Overlay Network Construction: Improved Overall and Node-Wise Message Complexity. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chang_et_al:LIPIcs.FSTTCS.2025.21,
  author =	{Chang, Yi-Jun and Chen, Yanyu and Mishra, Gopinath},
  title =	{{Overlay Network Construction: Improved Overall and Node-Wise Message Complexity}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{21:1--21:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.21},
  URN =		{urn:nbn:de:0030-drops-251025},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.21},
  annote =	{Keywords: Distributed algorithms, Overlay networks, Expander graphs}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.33
Edge Densities of Drawings of Graphs with One Forbidden Cell

Authors: Benedikt Hahn, Torsten Ueckerdt, and Birgit Vogtenhuber

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
A connected topological drawing of a graph divides the plane into a number of cells. The type of a cell c is the cyclic sequence of crossings and vertices along the boundary walk of c. For example, all triangular cells with three incident crossings and no incident vertex share the same cell type. When a non-homotopic drawing of an n-vertex multigraph G does not contain any such cells, Ackerman and Tardos [JCTA 2007] proved that G has at most 8n-20 edges, while Kaufmann, Klemz, Knorr, Reddy, Schröder, and Ueckerdt [GD 2024] showed that this bound is tight. In this paper, we initiate the in-depth study of non-homotopic drawings that do not contain one fixed cell type 𝔠, and investigate the edge density of the corresponding multigraphs, i.e., the maximum possible number of edges. We consider non-homotopic as well as simple drawings, multigraphs as well as simple graphs, and every possible type of cell. For every combination of drawing style, graph type, and cell type, we give upper and lower bounds on the corresponding edge density. With the exception of the cell type with four incident crossings and no incident vertex, we show for every cell type 𝔠 that the edge density of n-vertex (multi)graphs with 𝔠-free drawings is either quadratic in n or linear in n. In most cases, our bounds are tight up to an additive constant. Additionally, we improve the current lower bound on the edge density of simple graphs that admit a non-homotopic quasiplanar drawing from 7n-28 to 7.5n-28.
Cite as

Benedikt Hahn, Torsten Ueckerdt, and Birgit Vogtenhuber. Edge Densities of Drawings of Graphs with One Forbidden Cell. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hahn_et_al:LIPIcs.GD.2025.33,
  author =	{Hahn, Benedikt and Ueckerdt, Torsten and Vogtenhuber, Birgit},
  title =	{{Edge Densities of Drawings of Graphs with One Forbidden Cell}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{33:1--33:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.33},
  URN =		{urn:nbn:de:0030-drops-250199},
  doi =		{10.4230/LIPIcs.GD.2025.33},
  annote =	{Keywords: Edge density, cell types, forbidden substructures, non-homotopic drawings, simple drawings}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.22
Constructing Long Paths in Graph Streams

Authors: Christian Konrad and Chhaya Trehan

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

Abstract
In the graph stream model of computation, an algorithm processes the edges of an n-vertex input graph in one or more sequential passes while using a memory that is sublinear in the input size. The streaming model poses significant challenges for algorithmically constructing long paths. Many known algorithms that are tasked with extending an existing path as a subroutine require an entire pass over the input to add a single additional edge. This raises a fundamental question: Are multiple passes inherently necessary to construct paths of non-trivial lengths, or can a single pass suffice? To address this question, we systematically study the Longest Path problem in the one-pass streaming model. In this problem, given a desired approximation factor α, the objective is to compute a path of length at least lp(G)/α, where lp(G) is the length of a longest path in the input graph G. We study the problem in the insertion-only and the insertion-deletion streaming models, and we give algorithms as well as space lower bounds for both undirected and directed graphs. Our results are: 1) We show that for undirected graphs, in both the insertion-only and the insertion-deletion models, there are semi-streaming algorithms, i.e., algorithms that use space O(n poly log n), that compute a path of length at least d/3 with high probability, where d is the average degree of the input graph. These algorithms can also yield an α-approximation to Longest Path using space Õ(n²/α). 2) Next, we show that such a result cannot be achieved for directed graphs, even in the insertion-only model. We show that computing a (n^{1-o(1)})-approximation to Longest Path in directed graphs in the insertion-only model requires space Ω(n²). This result is in line with recent results that demonstrate that processing directed graphs is often significantly harder than undirected graphs in the streaming model. 3) We further complement our results with two additional lower bounds. First, we show that semi-streaming space is insufficient for small constant factor approximations to Longest Path for undirected graphs in the insertion-only model. Last, in undirected graphs in the insertion-deletion model, we show that computing an α-approximation requires space Ω(n²/α³).
Cite as

Christian Konrad and Chhaya Trehan. Constructing Long Paths in Graph Streams. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{konrad_et_al:LIPIcs.ESA.2025.22,
  author =	{Konrad, Christian and Trehan, Chhaya},
  title =	{{Constructing Long Paths in Graph Streams}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{22:1--22:19},
  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.2025.22},
  URN =		{urn:nbn:de:0030-drops-244902},
  doi =		{10.4230/LIPIcs.ESA.2025.22},
  annote =	{Keywords: Longest Path Problem, Streaming Algorithms, One-way Two-party Communication Complexity}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.60
Sink-Free Orientations: A Local Sampler with Applications

Authors: Konrad Anand, Graham Freifeld, Heng Guo, Chunyang Wang, and Jiaheng Wang

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

Abstract
For sink-free orientations in graphs of minimum degree at least 3, we show that there is a deterministic approximate counting algorithm that runs in time O((n^33/ε^32)log(n/ε)), a near-linear time sampling algorithm, and a randomised approximate counting algorithm that runs in time O((n/ε)²log(n/ε)), where n denotes the number of vertices of the input graph and 0 < ε < 1 is the desired accuracy. All three algorithms are based on a local implementation of the sink popping method (Cohn, Pemantle, and Propp, 2002) under the partial rejection sampling framework (Guo, Jerrum, and Liu, 2019).
Cite as

Konrad Anand, Graham Freifeld, Heng Guo, Chunyang Wang, and Jiaheng Wang. Sink-Free Orientations: A Local Sampler with Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 60:1-60:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{anand_et_al:LIPIcs.APPROX/RANDOM.2025.60,
  author =	{Anand, Konrad and Freifeld, Graham and Guo, Heng and Wang, Chunyang and Wang, Jiaheng},
  title =	{{Sink-Free Orientations: A Local Sampler with Applications}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{60:1--60:19},
  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.2025.60},
  URN =		{urn:nbn:de:0030-drops-244267},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.60},
  annote =	{Keywords: Sink-free orientations, local sampling, deterministic counting}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.2
Covering Simple Orthogonal Polygons with Rectangles

Authors: Aniket Basu Roy

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

Abstract
We study the problem of Covering Orthogonal Polygons with Rectangles, focusing on three variants: covering the interior, the boundary, and the corners. While previous work provided constant-factor approximation algorithms for these problems, significant improvements had not been achieved for over two decades. The main contribution of this work is the development of a Polynomial Time Approximation Scheme (PTAS) for both the Boundary Cover and Corner Cover problems on simple polygons, using a local search algorithm. Our work advances the state of the art, improving upon the previous best-known 4-approximation for the Boundary Cover and 2-approximation for the Corner Cover problems. The technical core of our work lies in proving the existence of planar support graphs for certain geometric hypergraphs defined by the polygon and its containment-maximal rectangles. This structural insight enables the application of the local search framework to achieve the PTAS results. We also demonstrate the limitations of this approach by constructing instances where local search fails for the Interior Cover and certain dual problems, such as the Maximum Antirectangle and Hitting Set problems. Additionally, the methods yield a PTAS for a special case of the Discrete Independent Set problem for rectangles. These results not only settle longstanding open questions but also introduce new techniques that may be of independent interest within computational geometry.
Cite as

Aniket Basu Roy. Covering Simple Orthogonal Polygons with Rectangles. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 2:1-2:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{basuroy:LIPIcs.APPROX/RANDOM.2025.2,
  author =	{Basu Roy, Aniket},
  title =	{{Covering Simple Orthogonal Polygons with Rectangles}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{2:1--2:23},
  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.2025.2},
  URN =		{urn:nbn:de:0030-drops-243686},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.2},
  annote =	{Keywords: Polygon Covering, Approximation Algorithms, Orthogonal Polygons, Rectangles, Local Search, Planar Supports}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.8
Multipass Linear Sketches for Geometric LP-Type Problems

Authors: N. Efe Çekirge, William Gay, and David P. Woodruff

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

Abstract
LP-type problems such as the Minimum Enclosing Ball (MEB), Linear Support Vector Machine (SVM), Linear Programming (LP), and Semidefinite Programming (SDP) are fundamental combinatorial optimization problems, with many important applications in machine learning applications such as classification, bioinformatics, and noisy learning. We study LP-type problems in several streaming and distributed big data models, giving ε-approximation linear sketching algorithms with a focus on the high accuracy regime with low dimensionality d, that is, when d < (1/ε)^0.999. Our main result is an O(ds) pass algorithm with O(s(√d/ε)^{3d/s}) ⋅ poly(d, log (1/ε)) space complexity in words, for any parameter s ∈ [1, d log (1/ε)], to solve ε-approximate LP-type problems of O(d) combinatorial and VC dimension. Notably, by taking s = d log (1/ε), we achieve space complexity polynomial in d and polylogarithmic in 1/ε, presenting exponential improvements in 1/ε over current algorithms. We complement our results by showing lower bounds of (1/ε)^Ω(d) for any 1-pass algorithm solving the (1 + ε)-approximation MEB and linear SVM problems, further motivating our multi-pass approach.
Cite as

N. Efe Çekirge, William Gay, and David P. Woodruff. Multipass Linear Sketches for Geometric LP-Type Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 8:1-8:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cekirge_et_al:LIPIcs.APPROX/RANDOM.2025.8,
  author =	{\c{C}ekirge, N. Efe and Gay, William and Woodruff, David P.},
  title =	{{Multipass Linear Sketches for Geometric LP-Type Problems}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{8:1--8:25},
  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.2025.8},
  URN =		{urn:nbn:de:0030-drops-243741},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.8},
  annote =	{Keywords: Streaming, sketching, LP-type problems}
}
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.2025.11
Crossing and Independent Families Among Polygons

Authors: Anna Brötzner, Robert Ganian, Thekla Hamm, Fabian Klute, and Irene Parada

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

Abstract
Given a set A of points in the plane, a family of line segments forming a matching in A is called crossing (or independent) if each pair of segments in the family intersects (or is non-intersecting, respectively). In past works, these notions have been generalized to polygons by identifying the points in A with the vertices of a given set of polygons and forbidding the line segments from intersecting or overlapping with polygon walls. In this work, we study the computational complexity of computing maximum crossing and independent families in this more general setting. As our first two results, we show that both problems are NP-hard already when the polygons are triangles. Motivated by this, we turn to parameterized algorithms. For our main algorithmic results, we consider the number of polygons on the input as the natural parameter and under this parameterization obtain a fixed-parameter algorithm for computing a largest crossing family among these polygons, and a separate XP-algorithm for computing a largest independent family that lies in one of the faces of the polygonal domain.
Cite as

Anna Brötzner, Robert Ganian, Thekla Hamm, Fabian Klute, and Irene Parada. Crossing and Independent Families Among Polygons. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{brotzner_et_al:LIPIcs.WADS.2025.11,
  author =	{Br\"{o}tzner, Anna and Ganian, Robert and Hamm, Thekla and Klute, Fabian and Parada, Irene},
  title =	{{Crossing and Independent Families Among Polygons}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.11},
  URN =		{urn:nbn:de:0030-drops-242424},
  doi =		{10.4230/LIPIcs.WADS.2025.11},
  annote =	{Keywords: crossing families, crossing-free matchings, segment intersection graphs, computational geometry, parameterized algorithms}
}
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