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Abstract

We study two related network design problems with two cost functions. In the buy-at-bulk k-Steiner tree problem we are given a graph G(V,E) with a set of terminals T ⊆ V including a particular vertex s called the root, and an integer k ≤ |T|. There are two cost functions on the edges of G, a buy cost \(b:E\longrightarrow {\mathbb{R}}^+\) and a distance cost \(r:E\longrightarrow {\mathbb{R}}^+\). The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost ∑\(_{e\in{\it H}}\) b(e)+∑\(_{t\in{\it T}-{\it s}}\) dist(t,s) is minimize, where dist(t,s) is the distance from t to s in H with respect to the r cost. We present an O(log4 n)-approximation for the buy-at-bulk k-Steiner tree problem. The second and closely related one is bicriteria approximation algorithm for Shallow-light k-Steiner trees. In the shallow-light k-Steiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e) over the edges, and an integer k. Our goal is to find a minimum cost (under b-cost) k-Steiner tree such that the diameter under r-cost is at most some given bound D. We develop an (O(logn),O(log3 n))-approximation algorithm for a relaxed version of Shallow-light k-Steiner tree where the solution has at least \(\frac{k}{8}\) terminals. Using this we obtain an (O(log2 n),O(log4 n))-approximation for the shallow-light k-Steiner tree and an O(log4 n)-approximation for the buy-at-bulk k-Steiner tree problem.

Keywords

Steiner Tree Network Design Problem Steiner Tree Problem Distance Cost Heavy Cluster 
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.
    Andrews, M.: Hardness of Buy-at-Bulk Network Design. In: Proceedings of FOCS 2004, pp. 115–124 (2004)Google Scholar
  2. 2.
    Andrews, M., Zhang, L.: Approximation algorithms for access network design. Algorithmica 34(2), 197–215 (2002)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Awerbuch, B., Azar, Y.: Buy-at-bulk network design. In: Proceedings of FOCS 1997, pp. 542–547 (1997)Google Scholar
  4. 4.
    Awerbuch, B., Azar, Y., Blum, A., Vempala, S.: New approximation guarantees for minimum-weight k-trees and prize-collecting salesmen. SIAM Journal on Computing 28(1), 254–262 (1999)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bar-Ilan, J., Kortsarz, G., Peleg, D.: Generalized submodular cover problems and applications. Theoretical Computer Science 250, 179–200 (2001)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bartal, Y.: On approximating arbitrary matrices by tree metrics. In: Proceedings of STOC, pp. 161–168 (1998)Google Scholar
  7. 7.
    Blum, A., Ravi, R., Vempala, S.: A constant-factor approximation algorithm for the k MST problem (extended abstract). In: Proceedings of STOC 1996, pp. 442–448 (1996)Google Scholar
  8. 8.
    Charikar, M., Karagiozova, A.: On non-uniform multicommodity buy-at-bulk network design. In: Proceedings of STOC 2005, pp. 176–182 (2005)Google Scholar
  9. 9.
    Chekuri, C., Khanna, S., Naor, J.: A deterministic algorithm for the cost-distance problem. In: Proceedings of SODA 2001, pp. 232–233 (2001)Google Scholar
  10. 10.
    Chuzhoy, J., Gupta, A., Naor, J., Sinha, A.: On the approximability of some network design problems. In: Proceedings of SODA 2005, pp. 943–951 (2005)Google Scholar
  11. 11.
    Chekuri, C., Hajiaghayi, M., Kortsarz, G., Salavatipour, M.: Approximation Algorithms for Non-Uniform Buy-at-Bulk Network Design Problems (submitted, 2006)Google Scholar
  12. 12.
    Cheriyan, J., Salman, F.S., Ravi, R., Subramanian, S.: Buy-at-bulk network design: Approximating the single-sink edge installation problem. SIAM Journal on Optimization 11(3), 595–610 (2000)MATHMathSciNetGoogle Scholar
  13. 13.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. Journal of Computer and System Sciences 69(3), 485–497 (2004)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Feige, U., Kortsarz, G., Peleg, D.: The dense k-subgraph problem. Algorithmica 29(3), 410–421 (2001)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Garg, N.: A 3-Approximation for the minimum tree spanning k vertices. In: Proceedings FOCS 1996, pp. 302–309 (1996)Google Scholar
  16. 16.
    Garg, N.: Saving an epsilon: a 2-approximation for the k-MST problem in graphs. In: Proceedings of STOC 2005, pp. 396–402 (2005)Google Scholar
  17. 17.
    Guha, S., Meyerson, A., Munagala, K.: A constant factor approximation for the single sink edge installation problems. In: Proceedings of STOC 2001, pp. 383–388 (2001)Google Scholar
  18. 18.
    Guha, S., Meyerson, A., Munagala, K.: Hierarchical placement and network design problems. In: Proceedings of FOCS 2001, pp. 603–612 (2001)Google Scholar
  19. 19.
    Gupta, A., Kumar, A., Pal, M., Roughgarden, T.: Approximation Via Cost-Sharing: A Simple Approximation Algorithm for the Multicommodity Rent-or-Buy Problem. In: Proceedings of FOCS 2003, pp. 606–617 (2003)Google Scholar
  20. 20.
    Gupta, A., Kumar, A., Roughgarden, T.: Simpler and better approximation algorithms for network design. In: Proceedings STOC 2003, pp. 365–372 (2003)Google Scholar
  21. 21.
    Hajiaghayi, M.T., Jain, K.: The Prize-Collecting Generalized Steiner Tree Problem via a new approach of Primal-Dual Schema. In: Proceedings of SODA 2006, pp. 631–640 (2006)Google Scholar
  22. 22.
    Hassin, R.: Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research 17(1), 36–42 (1992)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hassin, R., Levin, A.: Minimum Restricted Diameter Spanning trees. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 175–184. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  24. 24.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9, 256–278 (1974)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Kumar, A., Gupta, A., Roughgarden, T.: A Constant-Factor Approximation Algorithm for the Multicommodity Rent-or-Buy Problem. In: Proceedings of FOCS 2002, pp. 333–342 (2002)Google Scholar
  26. 26.
    Marathe, M., Ravi, R., Sundaram, R., Ravi, S.S., Rosenkrantz, D., Hunt, H.: Bicriteria network design problems. J. Algorithms 28(1), 141–171 (1998)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Meyerson, A., Munagala, K., Plotkin, S.: Cost-Distance: Two Metric Network Design. In: Proceedings of FOCS 2000, pp. 383–388 (2000)Google Scholar
  28. 28.
    Moss, A., Rabani, Y.: Approximation algorithms for constrained node weighted steiner tree problems. In: Proceedings of STOC 2001, pp. 373–382 (2001)Google Scholar
  29. 29.
    Ravi, R., Sundaram, R., Marathe, M.V., Rosenkrantz, D.J., Ravi, S.: Spanning trees short or small. SIAM Journal on Discrete Mathematics 9(2), 178–200 (1996)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • M. T. Hajiaghayi
    • 1
  • G. Kortsarz
    • 2
  • M. R. Salavatipour
    • 3
  1. 1.Department of Computer ScienceCarnegie Mellon University 
  2. 2.Department of Computer ScienceRutgers University-Camden 
  3. 3.Department of Computing ScienceUniversity of Alberta 

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