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Settling SETH vs. approximate sparse directed unweighted diameter (up to (NU)NSETH)

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Ray LiAuthors Info & Claims

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

Pages 1684 - 1696

https://doi.org/10.1145/3406325.3451045

Published: 15 June 2021 Publication History

Abstract

We prove several tight results on the fine-grained complexity of approximating the diameter of a graph. First, we prove that, for any ε>0, assuming the Strong Exponential Time Hypothesis (SETH), there are no near-linear time 2−ε-approximation algorithms for the Diameter of a sparse directed graph, even in unweighted graphs. This result shows that a simple near-linear time 2-approximation algorithm for Diameter is optimal under SETH, answering a question from a survey of Rubinstein and Vassilevska-Williams (SIGACT ’19) for the case of directed graphs.

In the same survey, Rubinstein and Vassilevska-Williams also asked if it is possible to show that there are no 2−ε approximation algorithms for Diameter in a directed graph in O(n^1.499) time. We show that, assuming a hypothesis called NSETH, one cannot use a deterministic SETH-based reduction to rule out the existence of such algorithms.

Extending the techniques in these two results, we characterize whether a 2−ε approximation algorithm running in time O(n^1+δ) for the Diameter of a sparse directed unweighted graph can be ruled out by a deterministic SETH-based reduction for every δ∈(0,1) and essentially every ε∈(0,1), assuming NSETH. This settles the SETH-hardness of approximating the diameter of sparse directed unweighted graphs for deterministic reductions, up to NSETH. We make the same characterization for randomized SETH-based reductions, assuming another hypothesis called NUNSETH.

We prove additional hardness and non-reducibility results for undirected graphs.

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Cited By

Bonnet É(2022)4 vs 7 Sparse Undirected Unweighted Diameter Is SETH-hard at Time n4/3ACM Transactions on Algorithms10.1145/349454018:2(1-14)Online publication date: 4-Mar-2022
https://dl.acm.org/doi/10.1145/3494540
Dalirrooyfard MLi RWilliams V(2022)Hardness of Approximate Diameter: Now for Undirected Graphs2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00102(1021-1032)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00102

Index Terms

Settling SETH vs. approximate sparse directed unweighted diameter (up to (NU)NSETH)
1. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness
  2. Design and analysis of algorithms
    1. Approximation algorithms analysis
    2. Graph algorithms analysis

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cover image ACM Conferences

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 2021

1797 pages

ISBN:9781450380539

DOI:10.1145/3406325

General Chair:
Samir Khuller
Northwestern University, USA
,
Program Chair:
Virginia Vassilevska Williams
Massachusetts Institute of Technology, USA

Copyright © 2021 ACM.

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Published: 15 June 2021

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STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing

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Cited By

Bonnet É(2022)4 vs 7 Sparse Undirected Unweighted Diameter Is SETH-hard at Time n4/3ACM Transactions on Algorithms10.1145/349454018:2(1-14)Online publication date: 4-Mar-2022
https://dl.acm.org/doi/10.1145/3494540
Dalirrooyfard MLi RWilliams V(2022)Hardness of Approximate Diameter: Now for Undirected Graphs2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00102(1021-1032)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00102

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