Internet and Network Economics

Volume 6484 of the series Lecture Notes in Computer Science pp 75-86

Specializations and Generalizations of the Stackelberg Minimum Spanning Tree Game

  • Davide BilòAffiliated withLancaster UniversityDipartimento di Informatica, Universit‘a dell’Aquila
  • , Luciano GualàAffiliated withLancaster UniversityDipartimento di Matematica, Universit‘a di Tor Vergata
  • , Stefano LeucciAffiliated withLancaster UniversityDipartimento di Matematica, Universit‘a di Tor Vergata
  • , Guido ProiettiAffiliated withLancaster UniversityDipartimento di Informatica, Universit‘a dell’AquilaIstituto di Analisi dei Sistemi ed Informatica, CNR

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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