Improved Approximations for Buy-at-Bulk and Shallow-Light k-Steiner Trees and (k,2)-Subgraph

  • M. Reza Khani
  • Mohammad R. Salavatipour
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)

Abstract

In this paper we give improved approximation algorithms for some network design problems. In the Bounded-Diameter or Shallow-Light k-Steiner tree problem (SLkST), we are given an undirected graph G = (V,E) with terminals T ⊆ V containing a root r ∈ T, a cost function c:E → ℝ + , a length function ℓ:E → ℝ + , a bound L > 0 and an integer k ≥ 1. The goal is to find a minimum c-cost r-rooted Steiner tree containing at least k terminals whose diameter under ℓ metric is at most L. The input to the Buy-at-Bulk k-Steiner tree problem (BBkST) is similar: graph G = (V,E), terminals T ⊆ V, cost and length functions c,ℓ:E → ℝ + , and an integer k ≥ 1. The goal is to find a minimum total cost r-rooted Steiner tree H containing at least k terminals, where the cost of each edge e is c(e) + ℓ(ef(e) where f(e) denotes the number of terminals whose path to root in H contains edge e. We present a bicriteria (O(log2n),O(logn))-approximation for SLkST: the algorithm finds a k-Steiner tree of diameter at most O(L·logn) whose cost is at most \(O(\log^2 n\cdot\mbox{\sc opt}^*)\) where \(\mbox{\sc opt}^*\) is the cost of an LP relaxation of the problem. This improves on the algorithm of [9] with ratio (O(log4n), O(log2n)). Using this, we obtain an O(log3n)-approximation for BBkST, which improves upon the O(log4n)-approximation of [9]. We also consider the problem of finding a minimum cost 2-edge-connected subgraph with at least k vertices, which is introduced as the (k,2)-subgraph problem in [14]. We give an O(logn)-approximation algorithm for this problem which improves upon the O(log2n)-approximation of [14].

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References

  1. 1.
    Andrews, M., Zhang, L.: Approximation algorithms for access network design. Algorithmica 32(2), 197–215 (2002); Preliminary version in Proc. of IEEE FOCS (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Awerbuch, B., Azar, Y.: Buy-at-bulk network design. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science, FOCS 1997 (1997)Google Scholar
  3. 3.
    Bateni, M., Chuzhoy, J.: Approximation Algorithms for the Directed k-Tour and k-stroll Problems. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302. Springer, Heidelberg (2010)Google Scholar
  4. 4.
    Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting High Log-Densities – an O(n 1/4)-Approximation for Densest k-Subgraph. In: Proceedings of Symposium on the Theory of Computing, STOC (2010)Google Scholar
  5. 5.
    Chekuri, C., Hajiaghayi, M., Kortsarz, G., Salavatipour, M.: Approximation Algorithms for Non-Uniform Buy-at-Bulk Network Design. SIAM J. on Computing 39(5), 1772–1798 (2009)CrossRefMATHGoogle Scholar
  6. 6.
    Chekuri, C., Khanna, S., Naor, J.: A Deterministic Approximation Algorithm for the Cost-Distance Problem Short paper. In: Proc. of ACM-SIAM SODA, pp. 232–233 (2001)Google Scholar
  7. 7.
    Garg, N.: Saving an epsilon: a 2-approximation for the k-MST problem in graphs. In: Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing (STOC), pp. 396–402 (2005)Google Scholar
  8. 8.
    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 the 44rd Symposium on Foundations of Computer Science, FOCS 2003 (2003)Google Scholar
  9. 9.
    Hajiaghayi, M.T., Kortsarz, G., Salavatipour, M.R.: Approximating buy-at-bulk and shallow-light k-steiner tree. Algorithmica 53(1), 89–103 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hassin, R.: Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research 17(1), 36–42 (1992)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khani, M.R., Salavatipour, M.R.: Approximation Algorithms for Min-max Tree Cover and Bounded Tree Cover Problems. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) RANDOM 2011 and APPROX 2011. LNCS, vol. 6845, pp. 302–314. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Kortsarz, G., Nutov, Z.: Approximating Some Network Design Problems with Node Costs. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 231–243. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Kumar, A., Gupta, A., Roughgarden, T.: A Constant-Factor Approximation Algorithm for the Multicommodity Rent-or-Buy Problem. In: Proceedings of FOCS 2002 (2002)Google Scholar
  14. 14.
    Lau, L., Naor, S., Salavatipour, M.R., Singh, M.: Survivable network design with degree or order constraints. SIAM J. on Computing 39(3), 1062–1087 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Marathe, M., Ravi, R., Sundaram, R., Ravi, S.S., Rosenkrantz, D., Hunt, H.B.: Bicriteria network design. Journal of Algorithms 28(1), 142–171 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Meyerson, A., Munagala, K., Plotkin, S.: Cost-Distance: Two Metric Network Design. SIAM J. on Computing, 2648–1659 (2008)Google Scholar
  17. 17.
    Ravi, R., Sundaram, R., Marathe, M.V., Rosenkrants, D.J., Ravi, S.S.: Spanning trees short or small. SIAM Journal on Discrete Mathematics 9(2), 178–200 (1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Salman, F.S., Cheriyan, J., Ravi, R., Subramanian, S.: Approximating the Single-Sink Link-Installation Problem in Network Design. SIAM J. on Optimization 11(3), 595–610 (2000)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Schrijver, A.: Combinatorial optimization: Polyhedra and Efficiency. Springer, Berlin (2003)MATHGoogle Scholar
  20. 20.
    Seymour, P.D.: In Graph Theory and related topics. In: Proc. Waterloo, pp. 341–355. Academic Press (1977/1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • M. Reza Khani
    • 1
  • Mohammad R. Salavatipour
    • 2
  1. 1.Dept. of Computing ScienceUniv. of AlbertaCanada
  2. 2.Toyota Tech. Inst. at Chicago, and Dept. of Computing ScienceUniv. of AlbertaCanada

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