Approximation Algorithms for Prize-Collecting Network Design Problems with General Connectivity Requirements

  • Chandrashekhar Nagarajan
  • Yogeshwer Sharma
  • David P. Williamson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5426)


In this paper, we introduce the study of prize-collecting network design problems having general connectivity requirements. Prior work considered only 0-1 or very limited connectivity requirements. We introduce general connectivity requirements in the prize-collecting generalized Steiner tree framework of Hajiaghayi and Jain [9], and consider penalty functions linear in the violation of the connectivity requirements. Using Jain’s iterated rounding algorithm [11] as a black box, and ideas from Goemans [7] and Levi, Lodi, Sviridenko [14], we give a 2.54-factor approximation algorithm for the problem. We also generalize the 0-1 requirements of PCF problem introduced by Sharma, Swamy, and Williamson [15] to include general connectivity requirements. Here we assume that the monotone submodular penalty function of Sharma et al. is generalized to a multiset function that can be decomposed into functions in the same form as that of Sharma et al. Using ideas from Goemans and Berstimas [6], we give an (αlogK)-approximation algorithm for the resulting problem, where K is the maximum connectivity requirement, and α= 2.54.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agrawal, A., Klein, P.N., Ravi, R.: When trees collide: An approximation algorithm for the generalized steiner problem on networks. SIAM J. Comput. 24(3), 440–456 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Balas, E.: The prize collecting traveling salesman problem. Networks 19, 621–636 (1989)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bienstock, D., Goemans, M.X., Simchi-Levi, D., Williamson, D.P.: A note on the prize collecting traveling salesman problem. Math. Programming 59, 413–420 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chimani, M., Kandyba, M., Mutzel, P.: A new ILP formulation for 2-root-connected prize-collecting steiner networks. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 681–692. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Feigenbaum, J., Papadimitriou, C.H., Shenker, S.: Sharing the cost of multicast transmissions. Journal of Computer and System Sciences 63, 21–41 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Goemans, M.X., Bertsimas, D.: Survivable networks, linear programming relaxations and the parsimonious property. Math. Program. 60, 145–166 (1993)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Goemans, M.: Personal communication (1998)Google Scholar
  8. 8.
    Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24(2), 296–317 (1995)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hajiaghayi, M.T., Jain, K.: The prize-collecting generalized Steiner tree problem via a new approach of primal-dual schema. In: SODA, pp. 631–640 (2006)Google Scholar
  10. 10.
    Hayrapetyan, A., Swamy, C., Tardos, É.: Network design for information networks. In: SODA, pp. 933–942 (2005)Google Scholar
  11. 11.
    Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Johnson, D.S., Minkoff, M., Phillips, S.: The prize collecting Steiner tree problem: theory and practice. In: SODA, pp. 760–769 (2000)Google Scholar
  13. 13.
    Kortsarz, G., Nutov, Z.: Approximating minimum cost connectivity problems. In: Gonzales, T. (ed.) Handbook of Approximation Algorithms and Metaheuristics. CRC Press, Boca Raton (2006)Google Scholar
  14. 14.
    Levi, R., Lodi, A., Sviridenko, M.I.: Approximation algorithms for the multi-item capacitated lot-sizing problem via flow-cover inequalities. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 454–468. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Sharma, Y., Swamy, C., Williamson, D.P.: Approximation algorithms for prize collecting forest problems with submodular penalty functions. In: SODA, pp. 1275–1284 (2007)Google Scholar
  16. 16.
    Williamson, D.P., Goemans, M.X., Mihail, M., Vazirani, V.V.: A primal-dual approximation algorithm for generalized steiner network problems. Combinatorica 15(3), 435–454 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Chandrashekhar Nagarajan
    • 1
  • Yogeshwer Sharma
    • 2
  • David P. Williamson
    • 1
  1. 1.School of OR&IECornell UniversityIthacaUSA
  2. 2.Department of Computer ScienceCornell UniversityIthacaUSA

Personalised recommendations