On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

  • Khaled Elbassioni
  • Rajiv Raman
  • Saurabh Ray
  • René Sitters
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5814)


In the tollbooth problem on trees, we are given a tree T= (V,E) with n edges, and a set of m customers, each of whom is interested in purchasing a path on the graph. Each customer has a fixed budget, and the objective is to price the edges of T such that the total revenue made by selling the paths to the customers that can afford them is maximized. An important special case of this problem, known as the highway problem, is when T is restricted to be a line. For the tollbooth problem, we present an O(logn)-approximation, improving on the current best O(logm)-approximation. We also study a special case of the tollbooth problem, when all the paths that customers are interested in purchasing go towards a fixed root of T. In this case, we present an algorithm that returns a (1 − ε)-approximation, for any ε> 0, and runs in quasi-polynomial time. On the other hand, we rule out the existence of an FPTAS by showing that even for the line case, the problem is strongly NP-hard. Finally, we show that in the discount model, when we allow some items to be priced below zero to improve the overall profit, the problem becomes even APX-hard.


Price Problem Price Function Discount Model Separator Node Variable Gadget 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aggarwal, G., Hartline, J.D.: Knapsack auctions. In: SODA 2006: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pp. 1083–1092. ACM Press, New York (2006)CrossRefGoogle Scholar
  2. 2.
    Balcan, M.F., Blum, A.: Approximation algorithms and online mechanisms for item pricing. In: EC 2006: Proceedings of the 7th ACM conference on Electronic commerce, pp. 29–35. ACM Press, New York (2006)CrossRefGoogle Scholar
  3. 3.
    Balcan, M.-F., Blum, A., Chan, H., Hajiaghayi, M.: A theory of loss-leaders: Making money by pricing below cost. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 293–299. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Balcan, M.F., Blum, A., Mansour, Y.: Item pricing for revenue maximization. In: EC 2008: Proceedings of the 9th ACM conference on Electronic commerce. ACM Press, New York (to appear) (2008)Google Scholar
  5. 5.
    Balcan, M.F., Blum, A.: Approximation algorithms and online mechanisms for item pricing. Theory of Computing 3, 179–195 (2007)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Briest, P.: Uniform budgets and the envy-free pricing problem. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 808–819. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Briest, P., Hoefer, M., Krysta, P.: Stackelberg network pricing games. In: Proc. 25th International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 133–142 (2008)Google Scholar
  8. 8.
    Briest, P., Krysta, P.: Single-minded unlimited supply pricing on sparse instances. In: SODA 2006: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pp. 1093–1102. ACM Press, New York (2006)CrossRefGoogle Scholar
  9. 9.
    Briest, P., Krysta, P.: Buying cheap is expensive: Hardness of non-parametric multi-product pricing. In: Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM-SIAM (2007)Google Scholar
  10. 10.
    Cheung, M., Swamy, C.: Approximation algorithms for single-minded envy-free profit-maximization problems with limited supply. In: FOCS 2008, pp. 35–44 (2008)Google Scholar
  11. 11.
    Demaine, E.D., Hajiaghayi, M.T., Feige, U., Salavatipour, M.R.: Combination can be hard: approximability of the unique coverage problem. In: SODA 2006: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pp. 162–171. ACM Press, New York (2006)CrossRefGoogle Scholar
  12. 12.
    Elbassioni, K.M., Sitters, R.A., Zhang, Y.: A quasi-PTAS for profit-maximizing pricing on line graphs. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 451–462. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Glynn, P.W., Van Roy, B., Rusmevichientong, P.: A nonparametric approach to multi-product pricing. Operations Research 54(1), 82–98 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Golovin, D., Nagarajan, V., Singh, M.: Approximating the k-multicut problem. In: SODA 2006, pp. 621–630 (2006)Google Scholar
  15. 15.
    Grigoriev, A., van Loon, J., Sitters, R., Uetz, M.: How to sell a graph: Guidelines for graph retailers. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 125–136. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Guruswami, V., Hartline, J.D., Karlin, A.R., Kempe, D., Kenyon, C., McSherry, F.: On profit-maximizing envy-free pricing. In: SODA 2005: Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, Philadelphia, PA, USA, pp. 1164–1173. Society for Industrial and Applied Mathematics (2005)Google Scholar
  17. 17.
    Hartline, J.D., Koltun, V.: Near-optimal pricing in near-linear time. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 422–431. Springer, Heidelberg (2005)Google Scholar
  18. 18.
    Khandekar, R., Kimbrel, T., Makarychev, K., Sviridenko, M.: On hardness of pricing items for single-minded bidders. In: Proceedings, 12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX (2009)Google Scholar
  19. 19.
    Luby, M., Wigderson, A.: Pairwise independence and derandomization. Foundations and Trends in Theoretical Computer Science 1(4), 237–301 (2005)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Motwani, R., Raghavan, P.: Randomized algorithms. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Khaled Elbassioni
    • 1
  • Rajiv Raman
    • 1
  • Saurabh Ray
    • 1
  • René Sitters
    • 2
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.Department of Econometrics and Operations ResearchVU UniversityAmsterdamthe Netherlands

Personalised recommendations