Improved Approximation Algorithms for Budgeted Allocations

  • Yossi Azar
  • Benjamin Birnbaum
  • Anna R. Karlin
  • Claire Mathieu
  • C. Thach Nguyen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)


We provide a 3/2-approximation algorithm for an offline budgeted allocations problem with applications to sponsored search auctions. This an improvement over the e/(e − 1) approximation of Andelman and Mansour [1] and the e/(e − 1) − ε approximation (for ε ≈ 0.0001) of Feige and Vondrak [2] for the more general Maximum Submodular Welfare (SMW) problem. For a special case of our problem, we improve this ratio to \(\sqrt{2}\). We also show that the problem is APX-hard.


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  1. 1.
    Andelman, N., Mansour, Y.: Auctions with Budget Constraints. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Feige, U., Vondrak, J.: Approximation algorithms for allocation problems: Improving the factor of 1 - 1/e. In: FOCS 2006, pp. 667–676 (2006)Google Scholar
  3. 3.
    Lahaie, S., Pennock, D., Saberi, A., Vohra, R.: Sponsored search auctions. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V. (eds.) Algorithmic Game Theory, pp. 699–716. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  4. 4.
    Buchbinder, N., Jain, K., Naor, J.: Online primal-dual algorithms for maximizing ad-auctions revenue. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 253–264. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: SODA 2008, pp. 982–991 (2008)Google Scholar
  6. 6.
    Mahdian, M., Nazerzadeh, H., Saberi, A.: Allocating online advertisement space with unreliable estimates. In: EC 2007, pp. 288–294 (2007)Google Scholar
  7. 7.
    Mehta, A., Saberi, A., Vazirani, U., Vazirani, V.: Adwords and generalized online matching. J. ACM 54(5), 22 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Lehmann, B., Lehmann, D., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. Games and Economic Behavior 55(2), 270–296 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chakrabarty, D., Goel, G.: On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP (manuscript, 2008)Google Scholar
  10. 10.
    Srinivasan, A.: Budgeted allocations in the full-information setting (manuscript, 2008)Google Scholar
  11. 11.
    Dobzinski, S., Schapira, M.: An improved approximation algorithm for combinatorial auctions with submodular bidders. In: SODA 2006, pp. 1064–1073 (2006)Google Scholar
  12. 12.
    Khot, S., Lipton, R., Markakis, E., Mehta, A.: Inapproximability Results for Combinatorial Auctions with Submodular Utility Functions. In: Deng, X., Ye, Y. (eds.) WINE 2005. LNCS, vol. 3828, pp. 92–101. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Mirrokni, V., Schapira, M., Vondrak, J.: Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions (manuscript, 2007)Google Scholar
  14. 14.
    Vondrak, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: STOC 2008 (to appear, 2008)Google Scholar
  15. 15.
    Fleischer, L., Goemans, M., Mirrokni, V., Sviridenko, M.: Tight approximation algorithms for maximum general assignment problems. In: SODA 2006, pp. 611–620 (2006)Google Scholar
  16. 16.
    Chekuri, C., Khanna, S.: A PTAS for the multiple knapsack problem. In: SODA 2000, pp. 213–222 (2000)Google Scholar
  17. 17.
    Shmoys, D., Tardos, E.: An approximation algorithm for the generalized assignment problem. Mathematical Programming 62, 461–474 (1993)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Andelman, N.: Online and strategic aspects of network resource management algorithms. PhD thesis, Tel Aviv University (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Yossi Azar
    • 1
  • Benjamin Birnbaum
    • 2
  • Anna R. Karlin
    • 2
  • Claire Mathieu
    • 3
  • C. Thach Nguyen
    • 2
  1. 1.Microsoft Research and Tel-Aviv UniversityUSA
  2. 2.University of WashingtonUSA
  3. 3.Brown UniversityUSA

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