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Matching Based Augmentations for Approximating Connectivity Problems

Part of the Lecture Notes in Computer Science book series (LNCS, volume 3887)

Abstract

We describe a very simple idea for designing approximation algorithms for connectivity problems: For a spanning tree problem, the idea is to start with the empty set of edges, and add matching paths between pairs of components in the current graph that have desirable properties in terms of the objective function of the spanning tree problem being solved. Such matching augment the solution by reducing the number of connected components to roughly half their original number, resulting in a logarithmic number of such matching iterations. A logarithmic performance ratio results for the problem by appropriately bounding the contribution of each matching to the objective function by that of an optimal solution.

In this survey, we trace the initial application of these ideas to traveling salesperson problems through a simple tree pairing observation down to more sophisticated applications for buy-at-bulk type network design problems.

Keywords

Span Tree Steiner Tree Performance Ratio Network Design Problem Span Tree Problem 
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.

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References

  1. 1.
    Chekuri, C., Khanna, S., Naor, S.: A deterministic algorithm for the COSTDISTANCE problem. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 232–233 (2001)Google Scholar
  2. 2.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Report 388, Graduate School of Industrial Administration, CMU (1976)Google Scholar
  3. 3.
    Frieze, A.M., Galbiati, G., Maffioli, F.: On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 12, 23–39 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fürer, M., Raghavachari, B.: An NC approximation algorithm for the minimumdegree spanning tree problem. In: Proceedings of the 28th Annual Allerton Conference on Communication, Control and Computing, pp. 274–281 (1990)Google Scholar
  5. 5.
    Goel, Estrin, D.: Simultaneous Optimization for Concave Costs: Single Sink Aggregation or Single Source Buy-at-Bulk. In: Proceedings of the 14th Annual ACMSIAM Symposium on Discrete Algorithms (2003)Google Scholar
  6. 6.
    Hochbaum, D. (ed.): Approximation algorithms for NP-hard problems. P.W.S (1997)Google Scholar
  7. 7.
    Klein, P.N., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms 19(1), 104–115 (1995); An early version appeared in the Proceedings of the Annual MPS conference on Integer Programming and Combinatorial Optimization (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Marathe, M.V., Ravi, R., Sundaram, R., Ravi, S.S., Rosenkrantz, D.J., Hunt, H.B.: Bicriteria network design problems. J. of Algorithms 28, 142–171 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Meyerson, A., Munagala, K., Plotkin, S.: Cost-Distance: Two Metric Network Design. In: Proceedings of the 41st Annual IEEE Symposium on the Foundations of Computer Science (2000)Google Scholar
  10. 10.
    Ravi, R., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Hunt III, H.B.: Many birds with one stone: Multi-objective approximation algorithms. In: Proceedings of the ACM Symposium on the Theory of Computing, pp. 438–447 (1993)Google Scholar
  11. 11.
    Ravi, R.: Rapid Rumor Ramification: Approximating the minimum broadcast time. In: Proceedings of the 35th Annual IEEE Symposium on the Foundations of Computer Science, pp. 202–213 (1994)Google Scholar
  12. 12.
    Ravi, R., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Hunt III, H.B.: Approximation Algorithms for Degree-Constrained Minimum-Cost Network- Design Problems. Algorithmica 31(1), 58–78 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • R. Ravi
    • 1
  1. 1.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA

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