Specializations and Generalizations of the Stackelberg Minimum Spanning Tree Game

  • Davide Bilò
  • Luciano Gualà
  • Stefano Leucci
  • Guido Proietti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)


The Stackelberg Minimum Spanning Tree (StackMST) game is a network pricing (bilevel) optimization problem. The game is played by two players on a graph G = (V,E), whose edges are partitioned into two sets: a set R of red edges (inducing a spanning tree of G) with a fixed non-negative real cost, and a set B of blue edges which are instead priced by a leader. This is done with the final intent of maximizing a revenue that will be returned for their purchase by a follower, whose goal in turn is to select a minimum spanning tree of G. StackMST is known to be APX-hard already when the number of distinct red costs is 2, as well as min {k, 1 + ln β, 1 + ln ρ}-approximable, where k is the number of distinct red costs, β is the number of blue edges selected by the follower in an optimal pricing, and ρ is the maximum ratio between red costs. In this paper we analyze some meaningful specializations and generalizations of StackMST, which shed some more light on the computational complexity of the game. More precisely, we first show that if G is complete, then the following holds: (i) if there are only 2 distinct red costs, then the problem can be solved optimally (this contrasts with the corresponding APX-hardness of the general problem); (ii) otherwise, the problem can be approximated within 7/4 + ε, for any ε> 0. Afterwards, we define a natural extension of StackMST, namely that in which blue edges have a non-negative activation cost associated, and the leader has a global activation budget that must not be exceeded, and, after showing that the very same approximation ratio as that of the original game can be achieved, we prove that if the spanning tree induced by the red edges has radius h (in terms of number of edges), then the problem admits a (2h + ε)-approximation algorithm.


Communication Networks Minimum Spanning Tree Stackelberg Games Network Pricing Games 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Biló, D., Gualà, L., Proietti, G., Widmayer, P.: Computational aspects of a 2-player Stackelberg shortest paths tree game. In: Papadimitriou, C., Zhang, S. (eds.) WINE 2008. LNCS, vol. 5385, pp. 251–262. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Biló, D., Gualà, L., Proietti, G.: Hardness of an asymmetric 2-player Stackelberg network pricing game. In: Electronic Colloquium on Computational Complexity (ECCC), TR09-112 (November 3, 2009)Google Scholar
  3. 3.
    Briest, P., Hoefer, M., Krysta, P.: Stackelberg network pricing games. In: Proc. of the 25th Ann. Symp. on Theoretical Aspects of Computer Science (STACS), pp. 133–142 (2008),
  4. 4.
    Briest, P., Hoefer, M., Gualà, L., 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
  5. 5.
    Cardinal, J., Demaine, E.D., Fiorini, S., Joret, G., Langerman, S., Newman, I., Weimann, O.: The Stackelberg minimum spanning tree game. Algorithmica (2009), doi:10.1007/s00453-009-9299-yGoogle Scholar
  6. 6.
    Cardinal, J., Demaine, E.D., 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
  7. 7.
    Grigoriev, A., van Hoesel, S., van der Kraaij, A., Uetz, M., Bouhtou, M.: Pricing network edges to cross a river. In: Persiano, G., Solis-Oba, R. (eds.) WAOA 2004. LNCS, vol. 3351, pp. 140–153. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Joret, G.: Stackelberg network pricing is hard to approximate (2009) (manuscript)Google Scholar
  9. 9.
    Labbé, M., Marcotte, P., Savard, G.: A bilevel model of taxation and its application to optimal highway pricing. Management Science 44(12), 608–622 (1998)zbMATHGoogle Scholar
  10. 10.
    Roch, S., Savard, G., Marcotte, P.: An approximation algorithm for Stackelberg network pricing. Networks 46(1), 57–67 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    van Hoesel, S.: An overview of Stackelberg pricing in networks, Research Memoranda 042, Maastricht: METEOR, Maastricht Research School of Economics of Technology and Organization (2006)Google Scholar
  12. 12.
    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

  • Davide Bilò
    • 1
  • Luciano Gualà
    • 2
  • Stefano Leucci
    • 2
  • Guido Proietti
    • 1
    • 3
  1. 1.Dipartimento di InformaticaUniversit‘a dell’AquilaL’AquilaItaly
  2. 2.Dipartimento di MatematicaUniversit‘a di Tor VergataRomaItaly
  3. 3.Istituto di Analisi dei Sistemi ed Informatica, CNRRomaItaly

Personalised recommendations