Improved Hardness of Approximation for Stackelberg Shortest-Path Pricing

  • Patrick Briest
  • Parinya Chalermsook
  • Sanjeev Khanna
  • Bundit Laekhanukit
  • Danupon Nanongkai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)


We consider the Stackelberg shortest-path pricing problem, which is defined as follows. Given a graph G with fixed-cost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapest s-t-path in G and we receive payment equal to the sum of prices of pricable edges belonging to the path. Our goal is to find prices maximizing the payment received from the customer. While Stackelberg shortest-path pricing was known to be APX-hard before, we provide the first explicit approximation threshold and prove hardness of approximation within 2 − o(1). We also argue that the nicely structured type of instance resulting from our reduction captures most of the challenges we face in dealing with the problem in general and, in particular, we show that the gap between the revenue of an optimal pricing and the only known general upper bound can still be logarithmically large.


Short Path Price Problem Valuation Function Cost Edge Approximation Guarantee 
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.
    Aggarwal, G., Feder, T., Motwani, R., Zhu, A.: Algorithms for multi-product pricing. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 72–83. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Arora, S., Lund, C.: Hardness Of Approximations. In: Approximation Algorithms for NP-hard Problems. PWS Publishing Company (1996)Google Scholar
  3. 3.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof Verification and Hardness of Approximation Problems. Journal of the ACM 45(3), 501–555 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balcan, N., Blum, A., Mansour, Y.: Item Pricing for Revenue Maximization. In: Proc. of 9th EC (2008)Google Scholar
  5. 5.
    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
  6. 6.
    Briest, P., Gualà, L., Hoefer, M., Ventre, C.: On Stackelberg Pricing with Computationally Bounded Consumers. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 42–54. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Briest, P., Hoefer, M., Krysta, P.: Stackelberg Network Pricing Games. In: Proc. of 25th STACS (2008)Google Scholar
  8. 8.
    Briest, P., Krysta, P.: Buying Cheap is Expensive: Hardness of Non-Parametric Multi-Product Pricing. In: Proc. of 18th SODA (2007)Google Scholar
  9. 9.
    Cardinal, J., Demaine, E., Fiorini, S., Joret, G., Langerman, S., Newman, I., Weimann, O.: The Stackelberg Minimum Spanning Tree Game. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 64–76. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Cardinal, J., Demaine, E., Fiorini, S., Joret, G., Newman, I., Weimann, O.: The Stackelberg Minimum Spanning Tree Game on Planar and Bounded-Treewidth Graphs. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 125–136. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Chuzhoy, J., Kannan, S., Khanna, S.: Network Pricing for Multicommodity Flows (2007) (unpublished manuscript)Google Scholar
  12. 12.
    Demaine, E.D., Feige, U., Hajiaghayi, M.T., Salavatipour, M.R.: Combination Can Be Hard: Approximability of the Unique Coverage Problem. In: Proc. of 17th SODA (2006)Google Scholar
  13. 13.
    Guruswami, V., Hartline, J.D., Karlin, A.R., Kempe, D., Kenyon, C., McSherry, F.: On Profit-Maximizing Envy-Free Pricing. In: Proc. of 16th SODA (2005)Google Scholar
  14. 14.
    Joret, G.: Stackelberg Network Pricing is Hard to Approximate. Networks n/a (2010), doi: 10.1002/net.20391Google Scholar
  15. 15.
    Raz, R.: A Parallel Repetition Theorem. SIAM Journal on Computing 27 (1998)Google Scholar
  16. 16.
    Roch, S., Savard, G., Marcotte, P.: An Approximation Algorithm for Stackelberg Network Pricing. Networks 46(1), 57–67 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    von Stackelberg, H.: Marktform und Gleichgewicht (Market and Equilibrium). Verlag von Julius Springer, Vienna (1934)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Patrick Briest
    • 1
  • Parinya Chalermsook
    • 2
  • Sanjeev Khanna
    • 3
  • Bundit Laekhanukit
    • 4
  • Danupon Nanongkai
    • 5
  1. 1.Department of Computer ScienceUniversity of PaderbornGermany
  2. 2.Department of Computer ScienceUniversity of ChicagoUSA
  3. 3.Department of Computer and Information ScienceUniversity of PennsylvaniaUSA
  4. 4.Department of Computer ScienceMcGill UniversityCanada
  5. 5.College of Computing, Georgia TechAtlantaUSA

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