research-article

An o(log² k)-approximation algorithm for the k-vertex connected spanning subgraph problem

Authors:

Jittat Fakcharoenphol,

Bundit LaekhanukitAuthors Info & Claims

STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing

Pages 153 - 158

https://doi.org/10.1145/1374376.1374401

Published: 17 May 2008 Publication History

Abstract

We present an O(log n• log k)-approximation algorithm for the problem of finding k-vertex connected spanning subgraph of minimum cost, where n is the number of vertices in the input graph, and k is the connectivity requirement. Our algorithm works for both directed and undirected graphs. The best known approximation guarantees for these problems are O(ln k• min{√k,n/n-k ln k}) by Kortsarz and Nutov, and O(ln{k}) in the case of undirected graphs where n≥ 6k² by Cheriyan, Vempala, and Vetta. Our algorithm is the first that has a polylogarithmic guarantee for all values of k. Combining our algorithm with the algorithm of Kortsarz and Nutov in case of small k, e.g., k<n/2, we have an O(log² k)-approximation algorithm. As in previous work, we use the Frank-Tardos algorithm for finding k-outconnected subgraphs as a subroutine. However, with a structural lemmas that we proved, we are able to show that we need only partial solutions returned by the Frank-Tardos algorithm; thus, we can avoid paying the whole cost of the optimal solution every time the algorithm is applied.

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Cited By

Chan PLau LSchild AWong SZhou H(2022)Network Design for s-t Effective ResistanceACM Transactions on Algorithms10.1145/352258818:3(1-45)Online publication date: 11-Oct-2022
https://dl.acm.org/doi/10.1145/3522588
Sadeghi EFan N(2019)On the survivable network design problem with mixed connectivity requirementsAnnals of Operations Research10.1007/s10479-019-03175-5Online publication date: 21-Feb-2019
https://doi.org/10.1007/s10479-019-03175-5
Georgiadis LItaliano GKaranasiou APapadopoulos CParotsidis N(2016)Sparse Subgraphs for 2-Connectivity in Directed GraphsProceedings of the 15th International Symposium on Experimental Algorithms - Volume 968510.1007/978-3-319-38851-9_11(150-166)Online publication date: 5-Jun-2016
https://dl.acm.org/doi/10.1007/978-3-319-38851-9_11
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Index Terms

An o(log² k)-approximation algorithm for the k-vertex connected spanning subgraph problem
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Network flows
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
      1. Network flows
  2. Randomness, geometry and discrete structures

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cover image ACM Conferences

STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing

May 2008

712 pages

ISBN:9781605580470

DOI:10.1145/1374376

General Chair:
Richard Ladner
University of Washington
,
Program Chair:
Cynthia Dwork
Microsoft Research, Silicon Valley

Copyright © 2008 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 17 May 2008

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May 17 - 20, 2008

British Columbia, Victoria, Canada

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Cited By

Chan PLau LSchild AWong SZhou H(2022)Network Design for s-t Effective ResistanceACM Transactions on Algorithms10.1145/352258818:3(1-45)Online publication date: 11-Oct-2022
https://dl.acm.org/doi/10.1145/3522588
Sadeghi EFan N(2019)On the survivable network design problem with mixed connectivity requirementsAnnals of Operations Research10.1007/s10479-019-03175-5Online publication date: 21-Feb-2019
https://doi.org/10.1007/s10479-019-03175-5
Georgiadis LItaliano GKaranasiou APapadopoulos CParotsidis N(2016)Sparse Subgraphs for 2-Connectivity in Directed GraphsProceedings of the 15th International Symposium on Experimental Algorithms - Volume 968510.1007/978-3-319-38851-9_11(150-166)Online publication date: 5-Jun-2016
https://dl.acm.org/doi/10.1007/978-3-319-38851-9_11
Nutov Z(2012)Approximating minimum-cost connectivity problems via uncrossable bifamiliesACM Transactions on Algorithms10.1145/2390176.23901779:1(1-16)Online publication date: 26-Dec-2012
https://dl.acm.org/doi/10.1145/2390176.2390177
Laekhanukit B(2011)An improved approximation algorithm for minimum-cost subset k-connectivityProceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I10.5555/2027127.2027130(13-24)Online publication date: 4-Jul-2011
https://dl.acm.org/doi/10.5555/2027127.2027130
Kamma LNutov Z(2011)Approximating Survivable Networks with Minimum Number of Steiner PointsApproximation and Online Algorithms10.1007/978-3-642-18318-8_14(154-165)Online publication date: 2011
https://doi.org/10.1007/978-3-642-18318-8_14
Gupta AKrishnaswamy RRavi RCharikar M(2010)Tree embeddings for two-edge-connected network designProceedings of the twenty-first annual ACM-SIAM symposium on Discrete algorithms10.5555/1873601.1873725(1521-1538)Online publication date: 17-Jan-2010
https://dl.acm.org/doi/10.5555/1873601.1873725
Nutov ZMathieu C(2009)An almost O(log k)-approximation for k-connected subgraphsProceedings of the twentieth annual ACM-SIAM symposium on Discrete algorithms10.5555/1496770.1496869(912-921)Online publication date: 4-Jan-2009
https://dl.acm.org/doi/10.5555/1496770.1496869
Gupta AKrishnaswamy RRavi RMitzenmacher M(2009)Online and stochastic survivable network designProceedings of the forty-first annual ACM symposium on Theory of computing10.1145/1536414.1536507(685-694)Online publication date: 31-May-2009
https://dl.acm.org/doi/10.1145/1536414.1536507
Chuzhoy JKhanna S(2009)An O(k^3 log n)-Approximation Algorithm for Vertex-Connectivity Survivable Network Design2009 50th Annual IEEE Symposium on Foundations of Computer Science10.1109/FOCS.2009.38(437-441)Online publication date: Oct-2009
https://doi.org/10.1109/FOCS.2009.38
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