research-article

Hardness of Distributed Optimization

Authors:

Keren Censor-Hillel,

Dean Leitersdorf,

Ami PazAuthors Info & Claims

PODC '19: Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

Pages 238 - 247

https://doi.org/10.1145/3293611.3331597

Published: 16 July 2019 Publication History

Abstract

This paper studies lower bounds for fundamental optimization problems in the CONGEST model. We show that solving problems exactly in this model can be a hard task, by providing tildeΩmega (n²) lower bounds for cornerstone problems, such as minimum dominating set (MDS), Hamiltonian path, Steiner tree and max-cut. These are almost tight, since all of these problems can be solved optimally in O(n²) rounds. Moreover, we show that even in bounded-degree graphs and even in simple graphs with maximum degree 5 and logarithmic diameter, it holds that various tasks, such as finding a maximum independent set (MaxIS) or a minimum vertex cover, are still difficult, requiring a near-tight number of tildeΩ (n) rounds.

Furthermore, we show that in some cases even approximations are difficult, by providing an tildeΩ (n²) lower bound for a (7/8+ε)-approximation for MaxIS, and a nearly-linear lower bound for an O(log n )-approximation for the k-MDS problem for any constant k geq≥ 2, as well as for several variants of the Steiner tree problem.

Our lower bounds are based on a rich variety of constructions that leverage novel observations, and reductions among problems that are specialized for the CONGEST model. However, for several additional approximation problems, as well as for exact computation of some central problems in P, such as maximum matching and max flow, we show that such constructions cannot be designed, by which we exemplify some limitations of this framework.

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Censor-Hillel KKhoury M(2024)On Distributed Computation of the Minimum Triangle Edge TransversalStructural Information and Communication Complexity10.1007/978-3-031-60603-8_19(336-358)Online publication date: 23-May-2024
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Index Terms

Hardness of Distributed Optimization
1. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

PODC '19: Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

July 2019

563 pages

ISBN:9781450362177

DOI:10.1145/3293611

General Chair:
Peter Robinson
City University of Hong Kong
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Program Chair:
Faith Ellen
University of Toronto, Canada

Copyright © 2019 ACM.

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Published: 16 July 2019

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PODC '19 Paper Acceptance Rate 48 of 173 submissions, 28%;

Overall Acceptance Rate 740 of 2,477 submissions, 30%

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Censor-Hillel KKhoury M(2024)On Distributed Computation of the Minimum Triangle Edge TransversalStructural Information and Communication Complexity10.1007/978-3-031-60603-8_19(336-358)Online publication date: 23-May-2024
https://doi.org/10.1007/978-3-031-60603-8_19
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Chang YLi ZOshman RNolin AHalldorsson MBalliu A(2023)The Complexity of Distributed Approximation of Packing and Covering Integer Linear ProgramsProceedings of the 2023 ACM Symposium on Principles of Distributed Computing10.1145/3583668.3594562(32-43)Online publication date: 19-Jun-2023
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KITAMURA NIZUMI T(2022)A Subquadratic-Time Distributed Algorithm for Exact Maximum MatchingIEICE Transactions on Information and Systems10.1587/transinf.2021EDP7083E105.D:3(634-645)Online publication date: 1-Mar-2022
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