DROPS

Document

DOI: 10.4230/LIPIcs.FSTTCS.2024.27

On Approximation Schemes for Stabbing Rectilinear Polygons

Authors: Arindam Khan, Aditya Subramanian, Tobias Widmann, and Andreas Wiese

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract

We study the problem of stabbing rectilinear polygons, where we are given n rectilinear polygons in the plane that we want to stab, i.e., we want to select horizontal line segments such that for each given rectilinear polygon there is a line segment that intersects two opposite (parallel) edges of it. Our goal is to find a set of line segments of minimum total length such that all polygons are stabbed. For the special case of rectangles, there is an O(1)-approximation algorithm and the problem is NP-hard [Chan, van Dijk, Fleszar, Spoerhase, and Wolff, 2018]. Also, the problem admits a QPTAS [Eisenbrand, Gallato, Svensson, and Venzin, 2021] and even a PTAS [Khan, Subramanian, and Wiese, 2022]. However, the approximability for the setting of more general polygons, e.g., L-shapes or T-shapes, is completely open. In this paper, we give conditions under which the problem admits a (1+ε)-approximation algorithm. We assume that each input polygon is composed of rectangles that are placed on top of each other. We show that if all input polygons satisfy the hourglass condition, then the problem admits a quasi-polynomial time approximation scheme. In particular, it is thus unlikely that this case is APX-hard. Furthermore, we show that there exists a PTAS if each input polygon is composed out of rectangles with a bounded range of widths. On the other hand, we prove that the general case of the problem (in which the input polygons may not satisfy these conditions) is APX-hard, already if all input polygons have only eight edges. We remark that all polygons with fewer edges automatically satisfy the hourglass condition. For arbitrary rectilinear polygons we even show a lower bound of Ω(log n) for the possible approximation ratio, which implies that the best possible ratio is in Θ(log n) since the problem is a special case of Set Cover.

Cite as

Arindam Khan, Aditya Subramanian, Tobias Widmann, and Andreas Wiese. On Approximation Schemes for Stabbing Rectilinear Polygons. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{khan_et_al:LIPIcs.FSTTCS.2024.27,
  author =	{Khan, Arindam and Subramanian, Aditya and Widmann, Tobias and Wiese, Andreas},
  title =	{{On Approximation Schemes for Stabbing Rectilinear Polygons}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{27:1--27:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.27},
  URN =		{urn:nbn:de:0030-drops-222166},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.27},
  annote =	{Keywords: Approximation Algorithms, Stabbing, Rectangles, Rectilinear Polygons, QPTAS, APX-hardness}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2023.46

Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set

Authors: Arindam Khan, Aditya Lonkar, Saladi Rahul, Aditya Subramanian, and Andreas Wiese

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

Abstract

Set cover and hitting set are fundamental problems in combinatorial optimization which are well-studied in the offline, online, and dynamic settings. We study the geometric versions of these problems and present new online and dynamic algorithms for them. In the online version of set cover (resp. hitting set), m sets (resp. n points) are given and n points (resp. m sets) arrive online, one-by-one. In the dynamic versions, points (resp. sets) can arrive as well as depart. Our goal is to maintain a set cover (resp. hitting set), minimizing the size of the computed solution. For online set cover for (axis-parallel) squares of arbitrary sizes, we present a tight O(log n)-competitive algorithm. In the same setting for hitting set, we provide a tight O(log N)-competitive algorithm, assuming that all points have integral coordinates in [0,N)². No online algorithm had been known for either of these settings, not even for unit squares (apart from the known online algorithms for arbitrary set systems). For both dynamic set cover and hitting set with d-dimensional hyperrectangles, we obtain (log m)^O(d)-approximation algorithms with (log m)^O(d) worst-case update time. This partially answers an open question posed by Chan et al. [SODA'22]. Previously, no dynamic algorithms with polylogarithmic update time were known even in the setting of squares (for either of these problems). Our main technical contributions are an extended quad-tree approach and a frequency reduction technique that reduces geometric set cover instances to instances of general set cover with bounded frequency.

Cite as

Arindam Khan, Aditya Lonkar, Saladi Rahul, Aditya Subramanian, and Andreas Wiese. Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 46:1-46:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{khan_et_al:LIPIcs.SoCG.2023.46,
  author =	{Khan, Arindam and Lonkar, Aditya and Rahul, Saladi and Subramanian, Aditya and Wiese, Andreas},
  title =	{{Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{46:1--46:17},
  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.2023.46},
  URN =		{urn:nbn:de:0030-drops-178967},
  doi =		{10.4230/LIPIcs.SoCG.2023.46},
  annote =	{Keywords: Geometric Set Cover, Hitting Set, Rectangles, Squares, Hyperrectangles, Online Algorithms, Dynamic Data Structures}
}

Search Results

Documents authored by Subramanian, Aditya

On Approximation Schemes for Stabbing Rectilinear Polygons

Abstract

Cite as

Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set

Abstract

Cite as

Thanks for your feedback!

Could not send message