Advertisement

Journal of Combinatorial Optimization

, Volume 31, Issue 2, pp 669–685 | Cite as

Improved approximations for buy-at-bulk and shallow-light \(k\)-Steiner trees and \((k,2)\)-subgraph

  • M. Reza Khani
  • Mohammad R. Salavatipour
Article
  • 103 Downloads

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 (SL\(k\)ST), we are given an undirected graph \(G=(V,E)\) with terminals \(T\subseteq V\) containing a root \(r\in T\), a cost function \(c:E\rightarrow \mathbb {R}^+\), a length function \(\ell :E\rightarrow \mathbb {R}^+\), a bound \(L>0\) and an integer \(k\ge 1\). The goal is to find a minimum \(c\)-cost \(r\)-rooted Steiner tree containing at least \(k\) terminals whose diameter under \(\ell \) metric is at most \(L\). The input to the buy-at-bulk \(k\)-Steiner tree problem (BB\(k\)ST) is similar: graph \(G=(V,E)\), terminals \(T\subseteq V\) containing a root \(r\in T\), cost and length functions \(c,\ell :E\rightarrow \mathbb {R}^+\), and an integer \(k\ge 1\). 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 \(c(e)+\ell (e)\cdot f(e)\) where \(f(e)\) denotes the number of terminals whose path to root in \(H\) contains edge \(e\). We present a bicriteria \((O(\log ^2 n),O(\log n))\)-approximation for SL\(k\)ST: the algorithm finds a \(k\)-Steiner tree with cost at most \(O(\log ^2 n\cdot \text{ opt }^*)\) where \(\text{ opt }^*\) is the cost of an LP relaxation of the problem and diameter at most \(O(L\cdot \log n)\). This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX’06/Algorithmica’09) which had ratio \((O(\log ^4 n), O(\log ^2 n))\). Using this, we obtain an \(O(\log ^3 n)\)-approximation for BB\(k\)ST, which improves upon the \(O(\log ^4 n)\)-approximation of Hajiaghayi et al. (2009). 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 Lau et al. (2009) (STOC’07/SICOMP09). This generalizes some well-studied classical problems such as the \(k\)-MST and the minimum cost \(2\)-edge-connected subgraph problems. We give an \(O(\log n)\)-approximation algorithm for this problem which improves upon the \(O(\log ^2 n)\)-approximation algorithm of Lau et al. (2009).

Keywords

Combinatorial optimization Approximation algorithms Network design Steiner tree \(k\)-edge connected 

Notes

Acknowledgments

We would like to thank an anonymous referee for her/his careful reading of this paper and the comments and suggestions.

References

  1. Andrews M (2004) Hardness of buy-at-bulk network design. Proceedings of IEEE FOCS, pp 115–124.Google Scholar
  2. Andrews M, Zhang L (2002) Approximation algorithms for access network design. Algorithmica 32(2):197–215CrossRefGoogle Scholar
  3. Antonakopoulos S, Chekuri C, Shepherd B, Zhang L (2011) Buy-at-bulk network design with protection. Mathe Oper Res 36(1):71–87MathSciNetCrossRefMATHGoogle Scholar
  4. Awerbuch B, Azar Y. (1997) Buy-at-bulk network design, In Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS ’97), pp 542–547.Google Scholar
  5. Awerbuch B, Azar Y, Blum A, Vempala S (1999) New approximation guarantees for minimum-weight \(k\)-trees and prize-collecting salesmen. SIAM J Comput 28(1):254–262MathSciNetCrossRefGoogle Scholar
  6. Bartal Y (1997) On approximating arbitrary metrics by tree metrics. In: Proceedings of ACM STOC, pp 161–168.Google Scholar
  7. Bar-Ilan J, Kortsarz G, Peleg D (2001) Generalized submodular cover problems and applications. Theor Comput Sci 250(1):179–200MathSciNetCrossRefMATHGoogle Scholar
  8. Bateni M, Chuzhoy J (2013) Approximation algorithms for the directed \(k\)-Tour and \(k\)-Stroll problems. Algorithmica 65(3):545–561MathSciNetCrossRefMATHGoogle Scholar
  9. Bhaskara A, Charikar M, Chlamtac E, Feige U, Vijayaraghavan A (2010) Detecting high log-densities: an \(O(n^{1/4})\)-approximation for densest \(k\)-subgraph. In: Proceedings of ACM Symposium on the Theory of Computing (STOC), pp 201–210.Google Scholar
  10. Blum A, Ravi R, Vempala S (1999) A constant-factor approximation algorithm for the \(k\)-MST problem. J Comput Syst Sci 58(1):101–108MathSciNetCrossRefMATHGoogle Scholar
  11. Chekuri C, Kumar A (2004) Maximum coverage problem with group budget constraints and applications. In: Approximation algorithms for combinatorial optimization, pp 72–83Google Scholar
  12. Chekuri C, Khanna S, Naor J (2001) A deterministic approximation algorithm for the cost-distance problem. Short paper in Proceedings of ACM-SIAM SODA, pp 232–233.Google Scholar
  13. Chekuri C, Hajiaghayi M, Kortsarz G, Salavatipour M (2006) Approximation algorithms for non-uniform buy-at-bulk network design problems. In: Proceedings of IEEE FOCS, pp 677–686.Google Scholar
  14. Chekuri C, Hajiaghayi M, Kortsarz G, Salavatipour M (2009) Approximation algorithms for non-uniform buy-at-bulk network design. SIAM J Comput 39(5):1772–1798MathSciNetCrossRefGoogle Scholar
  15. Chuzhoy J, Gupta A, Naor S, Sinha A (2005) On the approximability of some network design problems. Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pp 943–951. Society for Industrial and Applied Mathematics.Google Scholar
  16. Even G, Garg N, Konemann J, Ravi R, Sinha A (2004) Covering graphs using trees and stars. Oper Res Lett 32(4):309–315MathSciNetCrossRefMATHGoogle Scholar
  17. Fakcharoenphol J, Rao S, Talwar K (2004) A tight bound on approximating arbitrary metrics by tree metrics. J Comput Syst Sci 69(3):485–497MathSciNetCrossRefMATHGoogle Scholar
  18. Garg N (2005) 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.Google Scholar
  19. Guha S, Meyerson A, Munagala K (2009) A constant factor approximation for the single sink edge installation problem. SIAM J Comput 38(6):2426–2442Google Scholar
  20. Gupta A, Krishnaswamy R, Ravi R (2010) Tree embeddings for two-edge-connected network design. In: Proceedings of of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp 1521–1538, SODA.Google Scholar
  21. Gupta A, Kumar A, Pal M, Roughgarden T (2003) 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 ’03), pp 606–615. IEEE.Google Scholar
  22. Gupta A, Kumar A, Roughgarden T. Simpler and better approximation algorithms for network design, In Proceedings of the thirty-fifth ACM symposium on Theory of computing (STOC ’03), pp 365–372. ACM.Google Scholar
  23. Hajiaghayi MT, Kortsarz G, Salavatipour MR (2009) Approximating buy-at-bulk and shallow-light \(k\)-steiner tree. Algorithmica 53(1):89–103MathSciNetCrossRefMATHGoogle Scholar
  24. Hassin R (1992) Approximation schemes for the restricted shortest path problem. Math Oper Res 17(1):36–42MathSciNetCrossRefMATHGoogle Scholar
  25. Hassin R, Levin A (2002) Minimum restricted diameter spanning trees. Springer, Berlin, pp 175–184Google Scholar
  26. Jain K (2001) A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica 21(1):39–60MathSciNetCrossRefMATHGoogle Scholar
  27. Khani MR, Salavatipour MR (2011) Approximation algorithms for min-max tree cover and bounded tree cover problems. In: Proceedings of APPROX.Google Scholar
  28. Kortsarz G, Nutov Z (2009) Approximating some network design problems with vertex costs. In: Proceedings APPROX-RANDOM, pp 231–243.Google Scholar
  29. Kumar A, Gupta A, Roughgarden T (2002) A constant-factor approximation algorithm for the multicommodity rent-or-buy problem. In: Proceedings of the 43rd Symposium on Foundations of Computer Science (FOCS ’02), pp 333–342. IEEE.Google Scholar
  30. Lau L, Naor S, Salavatipour MR, Singh M (2009) Survivable network design with degree or order constraints. SIAM J Comput 39(3):1062–1087 Special issue for selected papers of STOC.Google Scholar
  31. Marathe MV, Ravi R, Sundaram R, Ravi SS, Rosenkrantz D, Hunt HB (1998) Bicriteria network design. J Algorithms 28(1):142–171Google Scholar
  32. Meyerson A, Munagala K, Plotkin S (2008) Cost-Distance: Two Metric Network Design. SIAM J. on Computing, 2648–1659.Google Scholar
  33. Ravi R, Sundaram R, Marathe MV, Rosenkrants DJ, Ravi SS (1996) Spanning trees short or small. SIAM J Discret Math 9(2):178–200CrossRefMATHGoogle Scholar
  34. Safari MA, Salavatipour MR (2011) A constant factor approximation for minimum \(\lambda \)-edge-connected \(k\)-subgraph with metric costs. SIAM J Discrete Math 25(3):1089–1102MathSciNetCrossRefMATHGoogle Scholar
  35. Salman FS, Cheriyan J, Ravi R, Subramanian S (2000) Approximating the single-sink link-installation problem in network design. SIAM J Optim 11(3):595–610MathSciNetCrossRefMATHGoogle Scholar
  36. Schrijver A (2003) Combinatorial optimization: Polyhedra and Efficiency. Springer-Verlag, BerlinGoogle Scholar
  37. Seymour PD (1981) Nowhere-zero 6-flows. J Comb Theory B 30(2):130–135Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of MarylandCollege parkUnited States
  2. 2.Department of Computing ScienceUniversity of AlbertaEdmontonCanada

Personalised recommendations