research-article

A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs

Authors:

Hans L. Bodlaender,

Sándor Kisfaludi-Bak,

Tom C. van der ZandenAuthors Info & Claims

STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

Pages 574 - 586

https://doi.org/10.1145/3188745.3188854

Published: 20 June 2018 Publication History

Abstract

We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to a wide range of geometric intersection graphs (intersections of similarly sized fat objects), yielding algorithms with running time 2^{O(n^1−1/d)} for any fixed dimension d≥ 2 for many well known graph problems, including Independent Set, r-Dominating Set for constant r, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms work on the graph itself, i.e., do not require any geometric information. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques.

The lower-bound framework is based on a constructive embedding of graphs into d-dimensional grids, and it allows us to derive matching 2^{Ω(n^1−1/d)} lower bounds under the Exponential Time Hypothesis even in the much more restricted class of d-dimensional induced grid graphs.

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Cited By

Bandyapadhyay SLochet WLokshtanov DSaurabh SXue J(2024)True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk GraphsACM Transactions on Algorithms10.1145/365604220:3(1-26)Online publication date: 8-Apr-2024
https://dl.acm.org/doi/10.1145/3656042
Bandyapadhyay SFriggstad ZMousavi R(2024)Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and VariantsAlgorithmica10.1007/s00453-024-01236-186:8(2557-2574)Online publication date: 13-May-2024
https://doi.org/10.1007/s00453-024-01236-1
Gupta SSa'ar GZehavi M(2023)Grid recognition: Classical and parameterized computational perspectivesJournal of Computer and System Sciences10.1016/j.jcss.2023.02.008136(17-62)Online publication date: Sep-2023
https://doi.org/10.1016/j.jcss.2023.02.008
Show More Cited By

Index Terms

A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs
1. Theory of computation
  1. Design and analysis of algorithms
    1. Parameterized complexity and exact algorithms
  2. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

June 2018

1332 pages

ISBN:9781450355599

DOI:10.1145/3188745

General Chairs:
Ilias Diakonikolas
University of Southern California, USA
,
David Kempe
University of Southern California, USA
,
Program Chair:
Monika Henzinger
University of Vienna, Austria

Copyright © 2018 ACM.

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Published: 20 June 2018

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STOC '18: Symposium on Theory of Computing

June 25 - 29, 2018

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Cited By

Bandyapadhyay SLochet WLokshtanov DSaurabh SXue J(2024)True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk GraphsACM Transactions on Algorithms10.1145/365604220:3(1-26)Online publication date: 8-Apr-2024
https://dl.acm.org/doi/10.1145/3656042
Bandyapadhyay SFriggstad ZMousavi R(2024)Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and VariantsAlgorithmica10.1007/s00453-024-01236-186:8(2557-2574)Online publication date: 13-May-2024
https://doi.org/10.1007/s00453-024-01236-1
Gupta SSa'ar GZehavi M(2023)Grid recognition: Classical and parameterized computational perspectivesJournal of Computer and System Sciences10.1016/j.jcss.2023.02.008136(17-62)Online publication date: Sep-2023
https://doi.org/10.1016/j.jcss.2023.02.008
Bhore SCarmi PKolay SZehavi M(2022)Parameterized Study of Steiner Tree on Unit Disk GraphsAlgorithmica10.1007/s00453-022-01020-z85:1(133-152)Online publication date: 5-Aug-2022
https://doi.org/10.1007/s00453-022-01020-z
An HGurumukhani MImpagliazzo RJaber MKünnemann MParga Nina M(2022)The Fine-Grained Complexity of Multi-Dimensional Ordering PropertiesAlgorithmica10.1007/s00453-022-01014-x84:11(3156-3191)Online publication date: 22-Aug-2022
https://doi.org/10.1007/s00453-022-01014-x
Cohen-Addad Vde Verdière ÉMarx Dde Mesmay A(2021)Almost Tight Lower Bounds for Hard Cutting Problems in Embedded GraphsJournal of the ACM10.1145/345070468:4(1-26)Online publication date: 14-Jul-2021
https://dl.acm.org/doi/10.1145/3450704
Wang HZhao Y(2021)Reverse Shortest Path Problem for Unit-Disk GraphsAlgorithms and Data Structures10.1007/978-3-030-83508-8_47(655-668)Online publication date: 31-Jul-2021
https://doi.org/10.1007/978-3-030-83508-8_47
de Berg MBodlaender HKisfaludi-Bak SMarx Dvan der Zanden T(2020)A Framework for Exponential-Time-Hypothesis--Tight Algorithms and Lower Bounds in Geometric Intersection GraphsSIAM Journal on Computing10.1137/20M132087049:6(1291-1331)Online publication date: 15-Dec-2020
https://doi.org/10.1137/20M1320870
Wang HXue J(2020)Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk GraphsDiscrete & Computational Geometry10.1007/s00454-020-00219-7Online publication date: 24-Jun-2020
https://doi.org/10.1007/s00454-020-00219-7
de Berg MKisfaludi-Bak S(2020)Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball GraphsTreewidth, Kernels, and Algorithms10.1007/978-3-030-42071-0_5(31-48)Online publication date: 20-Apr-2020
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