Stochastic Steiner Tree with Non-uniform Inflation

  • Anupam Gupta
  • MohammadTaghi Hajiaghayi
  • Amit Kumar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4627)


We study the Steiner Tree problem in the model of two-stage stochastic optimization with non-uniform inflation factors, and give a poly-logarithmic approximation factor for this problem. In this problem, we are given a graph G = (V,E), with each edge having two costs c M and c T (the costs for Monday and Tuesday, respectively). We are also given a probability distribution π: 2 V →[0,1] over subsets of V, and will be given a client set S drawn from this distribution on Tuesday. The algorithm has to buy a set of edges E M on Monday, and after the client set S is revealed on Tuesday, it has to buy a (possibly empty) set of edges E T (S) so that the edges in E M  ∪ E T (S) connect all the nodes in S. The goal is to minimize the c M (E M ) + E Sπ [ c T ( E T (S) ) ].

We give the first poly-logarithmic approximation algorithm for this problem. Our algorithm builds on the recent techniques developed by Chekuri et al. (FOCS 2006) for multi-commodity Cost-Distance. Previously, the problem had been studied for the cases when c T  = σ×c M for some constant σ ≥ 1 (i.e., the uniform case), or for the case when the goal was to find a tree spanning all the vertices but Tuesday’s costs were drawn from a given distribution \(\widehat{\pi}\) (the so-called “stochastic MST case”).

We complement our results by showing that our problem is at least as hard as the single-sink Cost-Distance problem (which is known to be Ω(loglogn) hard). Moreover, the requirement that Tuesday’s costs are fixed seems essential: if we allow Tuesday’s costs to dependent on the scenario as in stochastic MST, the problem becomes as hard as Label Cover (which is \(\Omega(2^{\log^{1-\varepsilon} n})\)-hard). As an aside, we also give an LP-rounding algorithm for the multi-commodity Cost-Distance problem, matching the O(log4 n) approximation guarantee given by Chekuri et al. (FOCS 2006).


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  1. 1.
    Alon, N., Karp, R.M., Peleg, D., West, D.: A graph-theoretic game and its application to the k-server problem. SIAM J. Comput. 24(1), 78–100 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Charikar, M., Chekuri, C., Pál, M.: Sampling bounds for stochastic optimization. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 257–269. Springer, Heidelberg (2005)Google Scholar
  3. 3.
    Charikar, M., Karagiozova, A.: On non-uniform multicommodity buy-at-bulk network design. In: STOC, pp. 176–182. ACM Press, New York (2005)Google Scholar
  4. 4.
    Chekuri, C., Hajiaghayi, M., Kortsarz, G., Salavatipour, M.R.: Approximation algorithms for non-uniform buy-at-bulk network design problems. In: FOCS (2006)Google Scholar
  5. 5.
    Chekuri, C., Khanna, S., Naor, J.S.: A deterministic algorithm for the cost-distance problem. In: SODA, pp. 232–233 (2001)Google Scholar
  6. 6.
    Chuzhoy, J., Gupta, A., Naor, J.S., Sinha, A.: On the approximability of network design problems. In: SODA, pp. 943–951 (2005)Google Scholar
  7. 7.
    Dhamdhere, K., Ravi, R., Singh, M.: On two-stage stochastic minimum spanning trees. In: Jünger, M., Kaibel, V. (eds.) Integer Programming and Combinatorial Optimization. LNCS, vol. 3509, pp. 321–334. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Elkin, M., Emek, Y., Spielman, D.A., Teng, S.-H.: Lower-stretch spanning trees. In: STOC, pp. 494–503. ACM Press, New York (2005)Google Scholar
  9. 9.
    Gupta, A., Pál, M.: Stochastic Steiner trees without a root. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1051–1063. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: Approximation algorithms for stochastic optimization problems. In: STOC, pp. 417–426 (2004)Google Scholar
  11. 11.
    Gupta, A., Pál, M., Ravi, R., Sinha, A.: What about Wednesday? approximation algorithms for multistage stochastic optimization. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 86–98. Springer, Heidelberg (2005)Google Scholar
  12. 12.
    Gupta, A., Ravi, R., Sinha, A.: An edge in time saves nine: LP rounding approximation algorithms for stochastic network design. In: FOCS, pp. 218–227 (2004)Google Scholar
  13. 13.
    Hajiaghayi, M.T., Kortsarz, G., Salavatipour, M.R.: Approximating buy-at-bulk k-steiner trees. In: Electronic Colloquium on Computational Complexity (ECCC) (2006)Google Scholar
  14. 14.
    Hajiaghayi, M.T., Kortsarz, G., Salavatipour, M.R.: Polylogarithmic approximation algorithm for non-uniform multicommodity buy-at-bulk. In: Electronic Colloquium on Computational Complexity (ECCC) (2006)Google Scholar
  15. 15.
    Hayrapetyan, A., Swamy, C., Tardos, E.: Network design for information networks. In: SODA, pp. 933–942 (2005)Google Scholar
  16. 16.
    Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.: On the costs and benefits of procrastination: Approximation algorithms for stochastic combinatorial optimization problems. In: SODA, pp. 684–693 (2004)Google Scholar
  17. 17.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. System Sci. 9, 256–278 (1974)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Klein, P., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms 19(1), 104–115 (1995)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Meyerson, A.: Online algorithms for network design. In: SPAA, pp. 275–280 (2004)Google Scholar
  20. 20.
    Meyerson, A., Munagala, K., Plotkin, S.: Cost-distance: Two metric network design. In: FOCS, pp. 624–630 (2000)Google Scholar
  21. 21.
    Ravi, R.: Matching based augmentations for approximating connectivity problems. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 13–24. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    Ravi, R., Sinha, A.: Hedging uncertainty: Approximation algorithms for stochastic optimization problems. In: IPCO, pp. 101–115 (2004)Google Scholar
  23. 23.
    Robins, G., Zelikovsky, A.: Tighter bounds for graph Steiner tree approximation. SIAM J. Discrete Math. 19(1), 122–134 (2005)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Shmoys, D., Swamy, C.: Stochastic optimization is (almost) as easy as deterministic optimization. In: FOCS, pp. 228–237 (2004)Google Scholar
  25. 25.
    Swamy, C., Shmoys, D.B.: Sampling-based approximation algorithms for multi-stage stochastic. In: FOCS, pp. 357–366 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Anupam Gupta
    • 1
  • MohammadTaghi Hajiaghayi
    • 2
  • Amit Kumar
    • 1
  1. 1.Computer Science Department, Carnegie Mellon University, Pittsburgh PA 15213USA
  2. 2.Department of Computer Science and Engineering, Indian Institute of Technology, New Delhi, 110016India

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