Skip to main content
SpringerLink
Log in

Sparse Subgraphs for 2-Connectivity in Directed Graphs

  • Conference paper
  • First Online:
Experimental Algorithms (SEA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9685))

Included in the following conference series:

  • 1056 Accesses

Abstract

Let G be a strongly connected directed graph. We consider the problem of computing the smallest strongly connected spanning subgraph of G that maintains the pairwise 2-vertex-connectivity of G, i.e., the 2-vertex-connected blocks of G (2VC-B). We provide linear-time approximation algorithms for this problem that achieve an approximation ratio of 6. Based on these algorithms, we show how to approximate, in linear time, within a factor of 6 the smallest strongly connected spanning subgraph of G that maintains respectively: both the 2-vertex-connected blocks and the 2-vertex-connected components of G (2VC-B-C); all the 2-connectivity relations of G (2C), i.e., both the 2-vertex- and the 2-edge-connected components and blocks. Moreover, we provide heuristics that improve the size of the computed subgraphs in practice, and conduct a thorough experimental study to assess their merits in practical scenarios.

G.F. Italiano and N. Parotsidis—Partially supported by MIUR under Project AMANDA.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Throughout, we use consistently the term bridge to refer to a bridge of a flow graph G(s) and the term strong bridge to refer to a strong bridge in the original graph G.

  2. 2.

    This follows from the fact that in the sparse subgraph the k vertices in blocks must have indegree at least two, while the remaining \(n-k\) vertices must have indegree at least one, since we seek for a strongly connected spanning subgraph.

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall Inc., Upper Saddle River (1993)

    MATH  Google Scholar 

  2. Alstrup, S., Harel, D., Lauridsen, P.W., Thorup, M.: Dominators in linear time. SIAM J. Comput. 28(6), 2117–2132 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buchsbaum, A.L., Georgiadis, L., Kaplan, H., Rogers, A., Tarjan, R.E., Westbrook, J.R.: Linear-time algorithms for dominators and other path-evaluation problems. SIAM J. Comput. 38(4), 1533–1573 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cheriyan, J., Thurimella, R.: Approximating minimum-size \(k\)-connected spanning subgraphs via matching. SIAM J. Comput. 30(2), 528–560 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Edmonds, J.: Edge-disjoint branchings. In: Rustin, B. (ed.) Combinatorial Algorithms, pp. 91–96. Academic Press, New York (1972)

    Google Scholar 

  6. Fakcharoenphol, J., Laekhanukit, B.: An \(o(\log ^2{k})\)-approximation algorithm for the \(k\)-vertex connected spanning subgraph problem. In: Proceedings of the 40th ACM Symposium on Theory of Computing, STOC 2008, pp. 153–158, New York, NY, USA, ACM (2008)

    Google Scholar 

  7. Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8, 399–404 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gabow, H.N., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. J. Comput. Syst. Sci. 30(2), 209–221 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  10. Georgiadis, L.: Approximating the smallest 2-vertex connected spanning subgraph of a directed graph. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 13–24. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  11. Georgiadis, L., Italiano, G.F., Laura, L., Parotsidis, N.: 2-Edge connectivity in directed graphs. SODA 2015, pp. 1988–2005 (2015)

    Google Scholar 

  12. Georgiadis, L., Italiano, G.F., Laura, L., Parotsidis, N.: 2-Vertex connectivity in directed graphs. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 605–616. Springer, Heidelberg (2015)

    Google Scholar 

  13. Georgiadis, L., Italiano, G.F., Papadopoulos, C., Parotsidis, N.: Approximating the smallest spanning subgraph for 2-edge-connectivity in directed graphs. In: Bansal, N., Finocchi, I. (eds.) Algorithms - ESA 2015. LNCS, vol. 9294, pp. 582–594. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48350-3_49

    Chapter  Google Scholar 

  14. Georgiadis, L., Italiano, G.F., Parotsidis, N.: A new framework for strong connectivity and 2-connectivity in directed graphs. CoRR, November 2015. arXiv:1511.02913

  15. Georgiadis, L., Tarjan, R.E.: Dominator tree certification and divergent spanning trees. ACM Trans. Algorithms 12(1), 11:1–11:42 (2015)

    Article  MathSciNet  Google Scholar 

  16. Henzinger, M., Krinninger, S., Loitzenbauer, V.: Finding 2-edge and 2-vertex strongly connected components in quadratic time. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 713–724. Springer, Heidelberg (2015)

    Google Scholar 

  17. Italiano, G.F., Laura, L., Santaroni, F.: Finding strong bridges and strong articulation points in linear time. Theor. Comput. Sci. 447, 74–84 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jaberi, R.: Computing the \(2\)-blocks of directed graphs. RAIRO-Theor. Inf. Appl. 49(2), 93–119 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jaberi, R.: On computing the 2-vertex-connected components of directed graphs. Discrete Applied Mathematics, (2015, to appear)

    Google Scholar 

  20. Khuller, S., Raghavachari, B., Young, N.E.: Approximating the minimum equivalent digraph. SIAM J. Comput. 24(4), 859–872 (1995). Announced at SODA 1994, 177–186

    Article  MathSciNet  MATH  Google Scholar 

  21. Khuller, S., Raghavachari, B., Young, N.E.: On strongly connected digraphs with bounded cycle length. Discrete Appl. Math. 69(3), 281–289 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kortsarz, G., Nutov, Z.: Approximating minimum cost connectivity problems. In: Gonzalez, T.F. (ed.) Approximation Algorithms and Metaheuristics. Chapman & Hall/CRC, Boca Raton (2007)

    Google Scholar 

  23. Laekhanukit, B., Oveis Gharan, S., Singh, M.: A rounding by sampling approach to the minimum size k-arc connected subgraph problem. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 606–616. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  24. Nagamochi, H., Ibaraki, T.: Algorithmic Aspects of Graph Connectivity, 1st edn. Cambridge University Press, New York (2008)

    Book  MATH  Google Scholar 

  25. Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tarjan, R.E.: Edge-disjoint spanning trees and depth-first search. Acta Informatica 6(2), 171–185 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vetta, A.: Approximating the minimum strongly connected subgraph via a matching lower bound. In: SODA, pp. 417–426 (2001)

    Google Scholar 

  28. Zhao, L., Nagamochi, H., Ibaraki, T.: A linear time 5/3-approximation for the minimum strongly-connected spanning subgraph problem. Inf. Process. Lett. 86(2), 63–70 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Ioannina, Ioannina, Greece

    Loukas Georgiadis, Aikaterini Karanasiou & Charis Papadopoulos

  2. Università di Roma “Tor Vergata”, Rome, Italy

    Giuseppe F. Italiano & Nikos Parotsidis

Authors
  1. Loukas Georgiadis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe F. Italiano
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aikaterini Karanasiou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Charis Papadopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Nikos Parotsidis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charis Papadopoulos .

Editor information

Editors and Affiliations

  1. Amazon.com, Inc., Palo Alto, California, USA

    Andrew V. Goldberg

  2. Russian Academy of Sciences, St. Petersburg, Russia

    Alexander S. Kulikov

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Georgiadis, L., Italiano, G.F., Karanasiou, A., Papadopoulos, C., Parotsidis, N. (2016). Sparse Subgraphs for 2-Connectivity in Directed Graphs. In: Goldberg, A., Kulikov, A. (eds) Experimental Algorithms. SEA 2016. Lecture Notes in Computer Science(), vol 9685. Springer, Cham. https://doi.org/10.1007/978-3-319-38851-9_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-38851-9_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-38850-2

  • Online ISBN: 978-3-319-38851-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation