research-article

A Framework for Exponential-Time-Hypothesis--Tight Algorithms and Lower Bounds in Geometric Intersection Graphs

Authors:

Hans L. Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx,

Tom C. van der ZandenAuthors Info & Claims

SIAM Journal on Computing, Volume 49, Issue 6

Pages 1291 - 1331

https://doi.org/10.1137/20M1320870

Published: 01 January 2020 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 intersection graphs of similarly-sized fat objects, yielding algorithms with running time $2^{O(n^{1-1/d})}$ for any fixed dimension $d\ge 2$ 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 are representation-agnostic, i.e., they work on the graph itself and do not require the geometric representation. 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 $2^{\Omega(n^{1-1/d})}$ lower bounds under the exponential time hypothesis even in the much more restricted class of $d$-dimensional induced grid graphs.

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cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 49, Issue 6

ISSN:0097-5397

DOI:10.1137/smjcat.49.6

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© 2020, Society for Industrial and Applied Mathematics.

This work is licensed under a Creative Commons Attribution 4.0 International License

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Published: 01 January 2020

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Cited By

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https://dl.acm.org/doi/10.1007/978-3-031-52113-3_29
de Berg MKisfaludi-Bak SMonemizadeh MTheocharous L(2023)Clique-Based Separators for Geometric Intersection GraphsAlgorithmica10.1007/s00453-022-01041-885:6(1652-1678)Online publication date: 1-Jun-2023
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https://dl.acm.org/doi/10.1007/978-3-031-38906-1_3
Kisfaludi-Bak SOkrasa KRzążewski P(2022)Computing List Homomorphisms in Geometric Intersection GraphsGraph-Theoretic Concepts in Computer Science10.1007/978-3-031-15914-5_23(313-327)Online publication date: 22-Jun-2022
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