research-article

Lower bounds for local approximation

Authors:

Jukka SuomelaAuthors Info & Claims

PODC '12: Proceedings of the 2012 ACM symposium on Principles of distributed computing

Pages 175 - 184

https://doi.org/10.1145/2332432.2332465

Published: 16 July 2012 Publication History

Abstract

In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient.

Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks.

As a corollary of our result and by prior work, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem.

Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is (α,r)-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius-r neighbourhoods. We show that there exists a finite (α,r)-homogeneous 2k-regular graph of girth at least g for any α<1 and any r, k, and g.

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Cited By

Hasemann HHirvonen JRybicki JSuomela J(2019)Deterministic local algorithms, unique identifiers, and fractional graph colouringTheoretical Computer Science10.1016/j.tcs.2014.06.044610:PB(204-217)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.1016/j.tcs.2014.06.044
Hella LJärvisalo MKuusisto ALaurinharju JLempiäinen TLuosto KSuomela JVirtema J(2018)Weak models of distributed computing, with connections to modal logicDistributed Computing10.1007/s00446-013-0202-328:1(31-53)Online publication date: 26-Dec-2018
https://dl.acm.org/doi/10.1007/s00446-013-0202-3
Göös MSuomela J(2018)No sublogarithmic-time approximation scheme for bipartite vertex coverDistributed Computing10.1007/s00446-013-0194-z27:6(435-443)Online publication date: 26-Dec-2018
https://dl.acm.org/doi/10.1007/s00446-013-0194-z
Show More Cited By

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Lower bounds for local approximation

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cover image ACM Conferences

PODC '12: Proceedings of the 2012 ACM symposium on Principles of distributed computing

July 2012

410 pages

ISBN:9781450314503

DOI:10.1145/2332432

General Chair:
Darek Kowalski
U. of Liverpool, UK and IMDEA Networks, Spain
,
Program Chair:
Alessandro Panconesi
Sapienza, University of Rome, Italy

Copyright © 2012 ACM.

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory
INESC-ID: INESC-ID Lisboa

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Published: 16 July 2012

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PODC '12: ACM Symposium on Principles of Distributed Computing

July 16 - 18, 2012

Madeira, Portugal

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

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Cited By

Hasemann HHirvonen JRybicki JSuomela J(2019)Deterministic local algorithms, unique identifiers, and fractional graph colouringTheoretical Computer Science10.1016/j.tcs.2014.06.044610:PB(204-217)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.1016/j.tcs.2014.06.044
Hella LJärvisalo MKuusisto ALaurinharju JLempiäinen TLuosto KSuomela JVirtema J(2018)Weak models of distributed computing, with connections to modal logicDistributed Computing10.1007/s00446-013-0202-328:1(31-53)Online publication date: 26-Dec-2018
https://dl.acm.org/doi/10.1007/s00446-013-0202-3
Göös MSuomela J(2018)No sublogarithmic-time approximation scheme for bipartite vertex coverDistributed Computing10.1007/s00446-013-0194-z27:6(435-443)Online publication date: 26-Dec-2018
https://dl.acm.org/doi/10.1007/s00446-013-0194-z
Mansour YPatt-Shamir BVardi S(2018)Constant-Time Local Computation AlgorithmsTheory of Computing Systems10.1007/s00224-017-9788-362:2(249-267)Online publication date: 1-Feb-2018
https://dl.acm.org/doi/10.1007/s00224-017-9788-3
IZUMI T(2015)Distributed Graph Algorithms and their Complexity: An IntroductionInterdisciplinary Information Sciences10.4036/iis.2015.L.0421:4(351-370)Online publication date: 2015
https://doi.org/10.4036/iis.2015.L.04
Göös MHirvonen JSuomela J(2013)Lower bounds for local approximationJournal of the ACM10.1145/252840560:5(1-23)Online publication date: 28-Oct-2013
https://dl.acm.org/doi/10.1145/2528405
Wawrzyniak WFatourou PTaubenfeld G(2013)Brief announcementProceedings of the 2013 ACM symposium on Principles of distributed computing10.1145/2484239.2484281(406-408)Online publication date: 22-Jul-2013
https://dl.acm.org/doi/10.1145/2484239.2484281
Fraigniaud PGöös MKorman ASuomela JFatourou PTaubenfeld G(2013)What can be decided locally without identifiers?Proceedings of the 2013 ACM symposium on Principles of distributed computing10.1145/2484239.2484264(157-165)Online publication date: 22-Jul-2013
https://dl.acm.org/doi/10.1145/2484239.2484264
Hella LJärvisalo MKuusisto ALaurinharju JLempiäinen TLuosto KSuomela JVirtema JKowalski DPanconesi A(2012)Weak models of distributed computing, with connections to modal logicProceedings of the 2012 ACM symposium on Principles of distributed computing10.1145/2332432.2332466(185-194)Online publication date: 16-Jul-2012
https://dl.acm.org/doi/10.1145/2332432.2332466
Hirvonen JSuomela JKowalski DPanconesi A(2012)Distributed maximal matchingProceedings of the 2012 ACM symposium on Principles of distributed computing10.1145/2332432.2332464(165-174)Online publication date: 16-Jul-2012
https://dl.acm.org/doi/10.1145/2332432.2332464
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