Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs

Reut Levi; Moti Medina; Omer Tubul

doi:10.4230/LIPIcs.APPROX/RANDOM.2024.60

Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs

Reut Levi, Moti Medina, Omer Tubul

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

In this paper, we study the problem of locally constructing a sparse spanning subgraph (LSSG), introduced by Levi, Ron, and Rubinfeld (ALGO’20). In this problem, the goal is to locally decide for each e ∈ E if it is in G^′ where G^′ is a connected subgraph of G (determined only by G and the randomness of the algorithm). We provide an LSSG that receives as a parameter a lower bound, ϕ, on the conductance of G whose query complexity is Õ(√n/ϕ²). This is almost optimal when ϕ is a constant since Ω(√n) queries are necessary even when G is an expander. Furthermore, this improves the state of the art of Õ(n²/³) queries for ϕ = Ω(1/n¹/¹²). We then extend our result for (k, ϕin, ϕout)-clusterable graphs and provide an algorithm whose query complexity is Õ(√n + ϕoutn) for constant k and ϕin. This bound is almost optimal when ϕout = O(1/√n).

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2024
Editors	Amit Kumar, Noga Ron-Zewi
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959773485
DOIs	https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.60
State	Published - Sep 2024
Event	27th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2024 and the 28th International Conference on Randomization and Computation, RANDOM 2024 - London, United Kingdom Duration: 28 Aug 2024 → 30 Aug 2024

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	317
ISSN (Print)	1868-8969

Conference

Conference	27th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2024 and the 28th International Conference on Randomization and Computation, RANDOM 2024
Country/Territory	United Kingdom
City	London
Period	28/08/24 → 30/08/24

Bibliographical note

Publisher Copyright:
© Reut Levi, Moti Medina, and Omer Tubul.

Keywords

Clusterbale Graphs
Locally Computable Algorithms
Spanning Subgraphs
Sublinear algorithms

Access to Document

10.4230/LIPIcs.APPROX/RANDOM.2024.60

Cite this

Levi, R., Medina, M., & Tubul, O. (2024). Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs. In A. Kumar, & N. Ron-Zewi (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2024 Article 60 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 317). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.60

Levi, Reut ; Medina, Moti ; Tubul, Omer. / Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2024. editor / Amit Kumar ; Noga Ron-Zewi. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2024. (Leibniz International Proceedings in Informatics, LIPIcs).

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abstract = "In this paper, we study the problem of locally constructing a sparse spanning subgraph (LSSG), introduced by Levi, Ron, and Rubinfeld (ALGO{\textquoteright}20). In this problem, the goal is to locally decide for each e ∈ E if it is in G′ where G′ is a connected subgraph of G (determined only by G and the randomness of the algorithm). We provide an LSSG that receives as a parameter a lower bound, ϕ, on the conductance of G whose query complexity is {\~O}(√n/ϕ2). This is almost optimal when ϕ is a constant since Ω(√n) queries are necessary even when G is an expander. Furthermore, this improves the state of the art of {\~O}(n2/3) queries for ϕ = Ω(1/n1/12). We then extend our result for (k, ϕin, ϕout)-clusterable graphs and provide an algorithm whose query complexity is {\~O}(√n + ϕoutn) for constant k and ϕin. This bound is almost optimal when ϕout = O(1/√n).",

keywords = "Clusterbale Graphs, Locally Computable Algorithms, Spanning Subgraphs, Sublinear algorithms",

author = "Reut Levi and Moti Medina and Omer Tubul",

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Levi, R, Medina, M & Tubul, O 2024, Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs. in A Kumar & N Ron-Zewi (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2024., 60, Leibniz International Proceedings in Informatics, LIPIcs, vol. 317, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 27th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2024 and the 28th International Conference on Randomization and Computation, RANDOM 2024, London, United Kingdom, 28/08/24. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.60

Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs. / Levi, Reut; Medina, Moti; Tubul, Omer.
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2024. ed. / Amit Kumar; Noga Ron-Zewi. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2024. 60 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 317).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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T1 - Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs

AU - Levi, Reut

AU - Medina, Moti

AU - Tubul, Omer

N1 - Publisher Copyright: © Reut Levi, Moti Medina, and Omer Tubul.

PY - 2024/9

Y1 - 2024/9

N2 - In this paper, we study the problem of locally constructing a sparse spanning subgraph (LSSG), introduced by Levi, Ron, and Rubinfeld (ALGO’20). In this problem, the goal is to locally decide for each e ∈ E if it is in G′ where G′ is a connected subgraph of G (determined only by G and the randomness of the algorithm). We provide an LSSG that receives as a parameter a lower bound, ϕ, on the conductance of G whose query complexity is Õ(√n/ϕ2). This is almost optimal when ϕ is a constant since Ω(√n) queries are necessary even when G is an expander. Furthermore, this improves the state of the art of Õ(n2/3) queries for ϕ = Ω(1/n1/12). We then extend our result for (k, ϕin, ϕout)-clusterable graphs and provide an algorithm whose query complexity is Õ(√n + ϕoutn) for constant k and ϕin. This bound is almost optimal when ϕout = O(1/√n).

AB - In this paper, we study the problem of locally constructing a sparse spanning subgraph (LSSG), introduced by Levi, Ron, and Rubinfeld (ALGO’20). In this problem, the goal is to locally decide for each e ∈ E if it is in G′ where G′ is a connected subgraph of G (determined only by G and the randomness of the algorithm). We provide an LSSG that receives as a parameter a lower bound, ϕ, on the conductance of G whose query complexity is Õ(√n/ϕ2). This is almost optimal when ϕ is a constant since Ω(√n) queries are necessary even when G is an expander. Furthermore, this improves the state of the art of Õ(n2/3) queries for ϕ = Ω(1/n1/12). We then extend our result for (k, ϕin, ϕout)-clusterable graphs and provide an algorithm whose query complexity is Õ(√n + ϕoutn) for constant k and ϕin. This bound is almost optimal when ϕout = O(1/√n).

KW - Clusterbale Graphs

KW - Locally Computable Algorithms

KW - Spanning Subgraphs

KW - Sublinear algorithms

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Levi R, Medina M, Tubul O. Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs. In Kumar A, Ron-Zewi N, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2024. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2024. 60. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.APPROX/RANDOM.2024.60

Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs

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Keywords

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