Advertisement

All-or-Nothing Generalized Assignment with Application to Scheduling Advertising Campaigns

  • Ron Adany
  • Moran Feldman
  • Elad Haramaty
  • Rohit Khandekar
  • Baruch Schieber
  • Roy Schwartz
  • Hadas Shachnai
  • Tami Tamir
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7801)

Abstract

We study a variant of the generalized assignment problem (gap) which we label all-or-nothing gap ( agap ). We are given a set of items, partitioned into n groups, and a set of m bins. Each item ℓ has size s  > 0, and utility a j  ≥ 0 if packed in bin j. Each bin can accommodate at most one item from each group, and the total size of the items in a bin cannot exceed its capacity. A group of items is satisfied if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total utility from satisfied groups is maximized. We motivate the study of agap by pointing out a central application in scheduling advertising campaigns.

Our main result is an O(1)-approximation algorithm for agap instances arising in practice, where each group consists of at most m/2 items. Our algorithm uses a novel reduction of agap to maximizing submodular function subject to a matroid constraint. For agap instances with fixed number of bins, we develop a randomized polynomial time approximation scheme (PTAS), relying on a non-trivial LP relaxation of the problem.

We present a (3 + ε)-approximation as well as PTASs for other special cases of agap, where the utility of any item does not depend on the bin in which it is packed. Finally, we derive hardness results for the different variants of agap studied in the paper.

Keywords

Total Size Generalize Assignment Polynomial Time Approximation Scheme Submodular Function Group Packing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adany, R., Feldman, M., Haramaty, E., Khandekar, R., Schieber, B., Schwartz, R., Shachnai, H., Tamir, T.: All-or-nothing generalized assignment with application to scheduling advertising campaigns (2012) Full version, http://www.cs.technion.ac.il/~hadas/PUB/AGAP_full.pdf
  2. 2.
    Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a submodular set function subject to a matroid constraint. SIAM J. on Computing 40(6), 1740–1766 (2011)zbMATHCrossRefGoogle Scholar
  3. 3.
    Chekuri, C., Khanna, S.: A PTAS for the multiple knapsack problem. SIAM J. on Computing 35(3), 713–728 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Dureau, V.: Addressable advertising on digital television. In: Proceedings of the 2nd European Conference on Interactive Television: Enhancing the Experience, Brighton, UK (March-April 2004)Google Scholar
  5. 5.
    Feige, U., Vondrák, J.: Approximation algorithms for allocation problems: Improving the factor of 1-1/e. In: FOCS, pp. 667–676 (2006)Google Scholar
  6. 6.
    Google AdWords, http://adwords.google.com
  7. 7.
    Kim, E.M., Wildman, S.S.: A deeper look at the economics of advertiser support for television: the implications of consumption-differentiated viewers and ad addressability. J. of Media Economics 19, 55–79 (2006)CrossRefGoogle Scholar
  8. 8.
    Kulik, A., Shachnai, H., Tamir, T.: Maximizing submodular set functions subject to multiple linear constraints. To appear in Mathematics of Operations ResearchGoogle Scholar
  9. 9.
    Kulik, A., Shachnai, H., Tamir, T.: Maximizing submodular set functions subject to multiple linear constraints. In: SODA, pp. 545–554 (2009)Google Scholar
  10. 10.
    Kundakcioglu, O.E., Alizamir, S.: Generalized assignment problem. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 1153–1162. Springer (2009)Google Scholar
  11. 11.
    Nemhauser, G., Wolsey, L., Fisher, M.: An analysis of the approximations for maximizing submodular set functions. Math. Programming 14, 265–294 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Nielsen Media Research. Advertising fact sheet. blog.nielsen.com (September 2010)Google Scholar
  13. 13.
    Nielsen Media Research. The cross-platform report, quarter 1, 2012 – US. blog.nielsen.com (May 2012)Google Scholar
  14. 14.
    Nielsen Media Research. Nielsen’s quarterly global adview pulse report. blog.nielsen.com (April 2012)Google Scholar
  15. 15.
    Nutov, Z., Beniaminy, I., Yuster, R.: A (1-1/e)-approximation algorithm for the generalized assignment problem. Operations Research Letters 34(3), 283–288 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Park, J., Lim, B., Lee, Y.: A Lagrangian dual-based branch-and-bound algorithm for the generalized multi-assignment problem. Management Science 44, 271–282 (1998)CrossRefGoogle Scholar
  17. 17.
    Pramataris, K., Papakyriakopoulos, D., Lekakos, G., Mulonopoulos, N.: Personalized Interactive TV Advertising: The iMEDIA Business Model. Electronic Markets 11, 1–9 (2001)CrossRefGoogle Scholar
  18. 18.
    Shmoys, D., Tardos, É.: An approximation algorithm for the generalized assignment problem. Mathematical Programming 62(1), 461–474 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
  20. 20.
    The Interactive Advertising Bureau (IAB), http://iab.net
  21. 21.
    Young, C.: Why TV spot length matters. Admap (497), 45–48 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ron Adany
    • 1
  • Moran Feldman
    • 2
  • Elad Haramaty
    • 2
  • Rohit Khandekar
    • 3
  • Baruch Schieber
    • 4
  • Roy Schwartz
    • 5
  • Hadas Shachnai
    • 2
  • Tami Tamir
    • 6
  1. 1.Computer Science DepartmentBar-Ilan UniversityRamat-GanIsrael
  2. 2.Computer Science DepartmentTechnionHaifaIsrael
  3. 3.Knight Capital GroupJersey CityUSA
  4. 4.IBM T.J. Watson Research CenterYorktown HeightsUSA
  5. 5.Microsoft ResearchRedmondUSA
  6. 6.School of Computer ScienceThe Interdisciplinary CenterHerzliyaIsrael

Personalised recommendations