Advertisement

Approximation Algorithms and Hardness Results for Labeled Connectivity Problems

  • Refael Hassin
  • Jérôme Monnot
  • Danny Segev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

Let G = (V,E) be a connected multigraph, whose edges are associated with labels specified by an integer-valued function \({\mathcal{L}} : E \rightarrow \mathbb{N}\). In addition, each label ℓ ∈ ℕ to which at least one edge is mapped has a non-negative cost c( ℓ). The minimum label spanning tree problem (MinLST) asks to find a spanning tree in G that minimizes the overall cost of the labels used by its edges. Equivalently, we aim at finding a minimum cost subset of labels I ⊆ ℕ such that the edge set \(\{ e \in E : {\mathcal {L}}( e ) \in I \}\) forms a connected subgraph spanning all vertices. Similarly, in the minimum label s -t path problem (MinLP) the goal is to identify an s-t path minimizing the combined cost of its labels, where s and t are provided as part of the input.

The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP. As a secondary objective, we make a concentrated effort to relate the algorithmic methods utilized in approximating these problems to a number of well-known techniques, originally studied in the context of integer covering.

Keywords

Vertex Cover Approximation Factor Hardness Result Connected Subgraph Approximation Guarantee 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ageev, A.A., Sviridenko, M.: Pipage rounding: A new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization 8(3), 307–328 (2004)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arora, S.: Personal communication (November 2005)Google Scholar
  3. 3.
    Arora, S., Sudan, M.: Improved low-degree testing and its applications. Combinatorica 23(3), 365–426 (2003)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Avidor, A., Zwick, U.: Approximating MIN 2-SAT and MIN 3-SAT. Theory of Computing Systems 38(3), 329–345 (2005)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bellare, M., Goldwasser, S., Lund, C., Russell, A.: Efficient probabilistically checkable proofs and applications to approximations. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pp. 294–304 (1993)Google Scholar
  6. 6.
    Broersma, H., Li, X., Woeginger, G., Zhang, S.: Paths and cycles in colored graphs. Australasian Journal on Combinatorics 31, 299–311 (2005)MATHMathSciNetGoogle Scholar
  7. 7.
    Brüggemann, T., Monnot, J., Woeginger, G.J.: Local search for the minimum label spanning tree problem with bounded color classes. Operations Research Letters 31(3), 195–201 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Carr, R.D., Doddi, S., Konjevod, G., Marathe, M.V.: On the red-blue set cover problem. In: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 345–353 (2000)Google Scholar
  9. 9.
    Chang, R.-S., Leu, S.-J.: The minimum labeling spanning trees. Information Processing Letters 63(5), 277–282 (1997)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Annals of Mathematics 162(1), 439–486 (2005)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goldschmidt, O., Hochbaum, D.S., Yu, G.: A modified greedy heuristic for the set covering problem with improved worst case bound. Information Processing Letters 48(6), 305–310 (1993)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hassin, R.: Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research 17(1), 36–42 (1992)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hassin, R., Monnot, J., Segev, D.: Approximation algorithms and hardness results for labeled connectivity problems (2006), available at http://www.math.tau.ac.il/~segevd
  14. 14.
    Karger, D.R., Motwani, R., Ramkumar, G.D.S.: On approximating the longest path in a graph. Algorithmica 18(1), 82–98 (1997)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Information Processing Letters 70(1), 39–45 (1999)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Krumke, S.O., Wirth, H.-C.: On the minimum label spanning tree problem. Information Processing Letters 66(2), 81–85 (1998)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lorenz, D.H., Raz, D.: A simple efficient approximation scheme for the restricted shortest path problem. Operations Research Letters 28(5), 213–219 (2001)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Marathe, M.V., Ravi, S.S.: On approximation algorithms for the minimum satisfiability problem. Information Processing Letters 58(1), 23–29 (1996)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pp. 475–484 (1997)Google Scholar
  20. 20.
    Srinivasan, A.: Distributions on level-sets with applications to approximation algorithms. In: Proceedings of the 42nd Annual Symposium on Foundations of Computer Science, pp. 588–597 (2001)Google Scholar
  21. 21.
    Wan, Y., Chen, G., Xu, Y.: A note on the minimum label spanning tree. Information Processing Letters 84(2), 99–101 (2002)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Wirth, H.-C.: Multicriteria Approximation of Network Design and Network Upgrade Problems. PhD thesis, Department of Computer Science, Würzburg University (2001)Google Scholar
  23. 23.
    Xiong, Y., Golden, B., Wasil, E.: Worst-case behavior of the MVCA heuristic for the minimum labeling spanning tree problem. Operations Research Letters 33(1), 77–80 (2005)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Refael Hassin
    • 1
  • Jérôme Monnot
    • 2
  • Danny Segev
    • 1
  1. 1.School of Mathematical SciencesTel-Aviv UniversityIsrael
  2. 2.CNRS LAMSADEUniversité Paris-DauphineFrance

Personalised recommendations