Unit Cost Buyback Problem

  • Yasushi Kawase
  • Xin Han
  • Kazuhisa Makino
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)


In this paper, we study unit cost buyback problem, i.e., the buyback problem with fixed cancellation cost for each cancelled item. The input is a sequence of elements e 1,e 2,…,e n , each of which has a weight w(e i ). We assume that weights have an upper and a lower bound, i.e., l ≤ w(e i ) ≤ u for any i. Given the ith element e i , we either accept e i or reject it with no cost, subject to some constraint on the set of accepted elements. In order to accept a new element e i , we could cancel some previous selected elements at a cost which is proportional to the number of elements canceled. Our goal is to maximize the profit, i.e., the sum of the weights of elements accepted (and not canceled) minus the total cancellation cost occurred. We construct optimal online algorithms and prove that they are the best possible, when the constraint is a matroid constraint or the unweighted knapsack constraint.


Knapsack Problem Competitive Ratio Online Algorithm Competitive Algorithm Knapsack Constraint 
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.


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  1. 1.
    Badanidiyuru Varadaraja, A.: Buyback problem - approximate matroid intersection with cancellation costs. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 379–390. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Ashwinkumar, B.V., Kleinberg, R.: Randomized online algorithms for the buyback problem. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 529–536. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Babaioff, M., Hartline, J.D., Kleinberg, R.D.: Selling banner ads: Online algorithms with buyback. In: Proceedings of 4th Workshop on Ad Auctions (2008)Google Scholar
  4. 4.
    Babaioff, M., Hartline, J.D., Kleinberg, R.D.: Selling ad campaigns: Online algorithms with cancellations. In: ACM Conference on Electronic Commerce, pp. 61–70 (2009)Google Scholar
  5. 5.
    Biyalogorsky, E., Carmon, Z., Fruchter, G.E., Gerstner, E.: Research note: Overselling with opportunistic cancellations. Marketing Science 18(4), 605–610 (1999)CrossRefGoogle Scholar
  6. 6.
    Constantin, F., Feldman, J., Muthukrishnan, S., Pál, M.: An online mechanism for ad slot reservations with cancellations. In: SODA, pp. 1265–1274 (2009)Google Scholar
  7. 7.
    Han, X., Kawase, Y., Makino, K.: Online knapsack problem with removal cost. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 61–73. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Han, X., Kawase, Y., Makino, K.: Randomized algorithms for removable online knapsack problems. In: Fellows, M., Tan, X., Zhu, B. (eds.) FAW-AAIM 2013. LNCS, vol. 7924, pp. 60–71. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Han, X., Makino, K.: Online minimization knapsack problem. In: Bampis, E., Jansen, K. (eds.) WAOA 2009. LNCS, vol. 5893, pp. 182–193. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Han, X., Makino, K.: Online removable knapsack with limited cuts. Theoretical Computer Science 411, 3956–3964 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Iwama, K., Taketomi, S.: Removable online knapsack problems. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 293–305. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Iwama, K., Zhang, G.: Optimal resource augmentations for online knapsack. In: APPROX-RANDOM, pp. 180–188 (2007)Google Scholar
  13. 13.
    Noga, J., Sarbua, V.: An online partially fractional knapsack problem. In: ISPAN, pp. 108–112 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Yasushi Kawase
    • 1
  • Xin Han
    • 2
  • Kazuhisa Makino
    • 3
  1. 1.University of TokyoJapan
  2. 2.Dalian University of TechnologyChina
  3. 3.Kyoto UniversityJapan

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