Skip to main content
SpringerLink
Log in

A Graph Reduction Step Preserving Element-Connectivity and Applications

  • Conference paper
Automata, Languages and Programming (ICALP 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5555))

Included in the following conference series:

Abstract

Given an undirected graph G = (V,E) and subset of terminals T ⊆ V, the element-connectivity κ′ G (u,v) of two terminals u,v ∈ T is the maximum number of u-v paths that are pairwise disjoint in both edges and non-terminals V ∖ T (the paths need not be disjoint in terminals). Element-connectivity is more general than edge-connectivity and less general than vertex-connectivity. Hind and Oellermann [18] gave a graph reduction step that preserves the global element-connectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise element-connectivities of the terminals.

We give two applications of the step to connectivity and network design problems: First, we show a polylogarithmic approximation for the problem of packing element-disjoint Steiner forests in general graphs, and an O(1)-approximation in planar graphs. Second, we find a very short and intuitive proof of a spider-decomposition theorem of Chuzhoy and Khanna [10] in the context of the single-sink k-vertex-connectivity problem. Our results highlight the effectiveness of the element-connectivity reduction step; we believe it will find more applications in the future.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aazami, A., Cheriyan, J., Jampani, K.R.: Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees. Algorithmica (manuscript submitted, 2008)

    Google Scholar 

  2. Calinescu, G., Chekuri, C., Vondrak, J.: Disjoint Bases in a Polymatroid. Random Structures & Algorithms (to appear)

    Google Scholar 

  3. Chakraborty, T., Chuzhoy, J., Khanna, S.: Network Design for Vertex Connectivity. In: Proc. of ACM STOC, pp. 167–176 (2008)

    Google Scholar 

  4. Chekuri, C., Korula, N.: Single-Sink Network Design with Vertex-Connectivity Requirements. In: Proc. of FST&TCS (2008)

    Google Scholar 

  5. Chekuri, C., Shepherd, B.: Approximate Integer Decompositions for Undirected Network Design Problems. SIAM J. on Disc. Math. 23(1), 163–177 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheriyan, J.: Personal Communication (October 2008)

    Google Scholar 

  7. Cheriyan, J., Salavatipour, M.: Hardness and Approximation Results for Packing Steiner Trees. Algorithmica 45(1), 21–43 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheriyan, J., Salavatipour, M.: Packing Element-disjoint Steiner Trees. ACM Trans. on Algorithms 3(4) (2007)

    Google Scholar 

  9. Cheriyan, J., Vempala, S., Vetta, A.: Network design via iterative rounding of setpair relaxations. Combinatorica 26(3), 255–275 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chuzhoy, J., Khanna, S.: Algorithms for Single-Source Vertex-Connectivity. In: Proc. of IEEE FOCS, pp. 105–114 (October 2008)

    Google Scholar 

  11. Chuzhoy, J., Khanna, S.: An O(k3 logn)-Approximation Algorithm for Vertex-Connectivity Survivable Network Design, arXiv.org preprint. arXiv:0812.4442v1

    Google Scholar 

  12. Feige, U., Halldórsson, M., Kortsarz, G., Srinivasan, A.: Approximating the Domatic Number. SIAM J. Comput. 32(1), 172–195 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fleischer, L., Jain, K., Williamson, D.P.: Iterative Rounding 2-approximation Algorithms for Minimum-cost Vertex Connectivity Problems. J. of Computer and System Sciences 72(5), 838–867 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Frank, A.: On Connectivity Properties of Eulerian Digraphs. Annals of Discrete Mathematics 41, 179–194 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Frank, A.: Augmenting graphs to meet edge-connectivity requirements. SIAM J. on Discrete Math. 5(1), 22–53 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Frank, A., Király, T., Kriesell, M.: On decomposing a hypergraph into k connected sub-hypergraphs. Discr. Applied Math. 131(2), 373–383 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grötschel, M., Martin, A., Weismantel, R.: The Steiner tree packing problem in VLSI-design. Math. Programming 78, 265–281 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hind, H.R., Oellermann, O.: Menger-type results for three or more vertices. Congressus Numerantium 113, 179–204 (1996)

    MathSciNet  MATH  Google Scholar 

  19. Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jain, K., Mahdian, M., Salavatipour, M.: Packing Steiner Trees. In: Proc. of ACM-SIAM SODA, pp. 266–274 (2003)

    Google Scholar 

  21. Jain, K., Mandoiu, I.I., Vazirani, V.V., Williamson, D.P.: A Primal-Dual Schema Based Approximation Algorithm for the Element Connectivity Problem. In: Proc. of ACM-SIAM SODA, pp. 484–489 (1999)

    Google Scholar 

  22. Jackson, B.: Some Remarks on Arc-connectivity, Vertex splitting, and Orientation in Digraphs. Journal of Graph Theory 12(3), 429–436 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Király, T., Lau, L.C.: Approximate Min-Max Theorems of Steiner Rooted-Orientations of Hypergraphs. J. of Combin. Theory B 98(6), 1233–1252 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kortsarz, G., Krauthgamer, R., Lee, J.R.: Hardness of Approximation for Vertex-Connectivity Network Design Problems. SIAM J. Comput. 33(3), 704–720 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kortsarz, G., Nutov, Z.: Approximating min-cost connectivity problems. In: Handbook on Approx. Algorithms and Metaheuristics, CRC Press, Boca Raton (2006)

    Google Scholar 

  26. Kriesell, M.: Edge-disjoint trees containing some given vertices in a graph. J. of Combin. Theory B 88, 53–63 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lau, L.C.: An approximate max-Steiner-tree-packing min-Steiner-cut theorem. Combinatorica 27(1), 71–90 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lau, L.C.: Packing steiner forests. In: Jünger, M., Kaibel, V. (eds.) IPCO 2005. LNCS, vol. 3509, pp. 362–376. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  29. Lovász, L.: On some connectivity properties of Eulerian graphs. Acta Math. Hung. 28, 129–138 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mader, W.: A reduction method for edge connectivity in graphs. Ann. Discrete Math. 3, 145–164 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nutov, Z.: An almost O(logk)-approximation for k-connected subgraphs. In: Proc. of ACM-SIAM SODA, pp. 912–921 (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Computer Science, University of Illinois, Urbana, IL, 61801, USA

    Chandra Chekuri & Nitish Korula

Authors
  1. Chandra Chekuri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nitish Korula
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Freiburg, Georges Köhler Allee 79, 79110, Freiburg, Germany

    Susanne Albers

  2. Department of Computer and Systems Sciences, Sapienza University of Rome, Via Ariosto 25, 00184, Roma, Italy

    Alberto Marchetti-Spaccamela

  3. Google R&D Center, School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Yossi Matias

  4. University of Patras and CTI, N. Kazantzaki Street 1, 26504 Rion, Patras, Greece

    Sotiris Nikoletseas

  5. RWTH Aachen, Lehrstuhl Informatik 7, Ahornstraße 55, 52056, Aachen, Germany

    Wolfgang Thomas

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chekuri, C., Korula, N. (2009). A Graph Reduction Step Preserving Element-Connectivity and Applications. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5555. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02927-1_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02927-1_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02926-4

  • Online ISBN: 978-3-642-02927-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation