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Documents authored by Singh, Satyam

Document
DOI: 10.4230/LIPIcs.SoCG.2026.64
Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree

Authors: Sándor Kisfaludi-Bak, Saeed Odak, Satyam Singh, and Geert van Wordragen

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

Abstract
We give an approximation scheme for the TSP in d-dimensional hyperbolic space that has optimal dependence on ε under Gap-ETH. For any fixed dimension d ≥ 2 and for any ε > 0 our randomized algorithm gives a (1+ε)-approximation in time 2^O(1/ε^{d-1}) n^{1+o(1)}. We also provide an algorithm for the hyperbolic Steiner tree problem with the same running time. Our algorithm is an Arora-style dynamic program based on a randomly shifted hierarchical decomposition. However, we introduce a new hierarchical decomposition called the hybrid hyperbolic quadtree to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25). Moreover, we have a new non-uniform portal placement, and our structure theorem employs a new weighted crossing analysis. We believe that these techniques could form the basis for further developments in geometric optimization in curved spaces.
Cite as

Sándor Kisfaludi-Bak, Saeed Odak, Satyam Singh, and Geert van Wordragen. Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 64:1-64:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kisfaludibak_et_al:LIPIcs.SoCG.2026.64,
  author =	{Kisfaludi-Bak, S\'{a}ndor and Odak, Saeed and Singh, Satyam and van Wordragen, Geert},
  title =	{{Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{64:1--64:17},
  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.2026.64},
  URN =		{urn:nbn:de:0030-drops-258710},
  doi =		{10.4230/LIPIcs.SoCG.2026.64},
  annote =	{Keywords: Hyperbolic traveling salesman problem, TSP, Hyperbolic Steiner tree problem, Approximation scheme, Banyan, Hyperbolic geometry}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.50
Online Hitting Sets for Disks of Bounded Radii

Authors: Minati De, Satyam Singh, and Csaba D. Tóth

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

Abstract
We present algorithms for the online minimum hitting set problem in geometric range spaces: Given a set P of n points in the plane and a sequence of geometric objects that arrive one-by-one, we need to maintain a hitting set at all times. For disks of radii in the interval [1,M], we present an O(log M log n)-competitive algorithm. This result generalizes from disks to positive homothets of any convex body in the plane with scaling factors in the interval [1,M]. As a main technical tool, we reduce the problem to the online hitting set problem for a finite subset of integer points and bottomless rectangles. Specifically, for a given N > 1, we present an O(log N)-competitive algorithm for the variant where P is a subset of an N× N section of the integer lattice, and the geometric objects are bottomless rectangles.
Cite as

Minati De, Satyam Singh, and Csaba D. Tóth. Online Hitting Sets for Disks of Bounded Radii. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{de_et_al:LIPIcs.ESA.2025.50,
  author =	{De, Minati and Singh, Satyam and T\'{o}th, Csaba D.},
  title =	{{Online Hitting Sets for Disks of Bounded Radii}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{50:1--50:16},
  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.2025.50},
  URN =		{urn:nbn:de:0030-drops-245181},
  doi =		{10.4230/LIPIcs.ESA.2025.50},
  annote =	{Keywords: Geometric Hitting Set, Online Algorithm, Homothets, Disks}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2022.17
Online Piercing of Geometric Objects

Authors: Minati De, Saksham Jain, Sarat Varma Kallepalli, and Satyam Singh

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract
We consider the online version of the piercing set problem where geometric objects arrive one by one. The online algorithm must maintain a piercing set for the arrived objects by making irrevocable decisions. First, we show that any deterministic online algorithm that solves this problem has a competitive ratio of at least Ω(n), which even holds when the objects are one-dimensional intervals. On the other hand, piercing unit objects is equivalent to the unit covering problem which is well-studied in the online model. Due to this, all the results related to the online unit covering problem are preserved for the online unit piercing problem when the objects are translated from each other. Surprisingly, no upper bound was known for the unit covering problem when unit objects are anything other than balls and hypercubes. In this paper, we introduce the notion of α-aspect and α-aspect_∞ objects. We give an upper bound of competitive ratio for α-aspect and α-aspect_∞ objects in ℝ³ and ℝ^d, respectively, with a scaling factor in the range [1,k]. We also propose a lower bound of the competitive ratio for bounded scaled objects like α-aspect objects in ℝ², axis-aligned hypercubes in ℝ^d, and balls in ℝ² and ℝ³. For piercing α-aspect_∞ objects in ℝ^d, we show that a simple deterministic algorithm achieves a competitive ratio of at most (2/α)^d((1+α)^d-1) (⌈log_(1+α)(2k/α)⌉)+1. This result is very general in nature. One can obtain upper bounds for specific objects by specifying the value of α. By putting the value of k = 1 to the above result, we get an upper bound of the competitive ratio for the unit covering problem for various types of objects.
Cite as

Minati De, Saksham Jain, Sarat Varma Kallepalli, and Satyam Singh. Online Piercing of Geometric Objects. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{de_et_al:LIPIcs.FSTTCS.2022.17,
  author =	{De, Minati and Jain, Saksham and Kallepalli, Sarat Varma and Singh, Satyam},
  title =	{{Online Piercing of Geometric Objects}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{17:1--17:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.17},
  URN =		{urn:nbn:de:0030-drops-174090},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.17},
  annote =	{Keywords: piercing set problem, online algorithm, competitive ratio, unit covering problem, geometric objects}
}
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