## The Stackelberg Minimum Spanning Tree Game (2007)

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Venue: | In Proc. of 10th WADS |

Citations: | 10 - 1 self |

### BibTeX

@INPROCEEDINGS{Cardinal07thestackelberg,

author = {Jean Cardinal and Erik D. Demaine and Samuel Fiorini and Gwenaël Joret and Stefan Langerman and Ilan Newman and Oren Weimann},

title = {The Stackelberg Minimum Spanning Tree Game},

booktitle = {In Proc. of 10th WADS},

year = {2007}

}

### OpenURL

### Abstract

Abstract. We consider a one-round two-player network pricing game, the Stackelberg Minimum Spanning Tree game or StackMST. The game is played on a graph (representing a network), whose edges are colored either red or blue, and where the red edges have a given fixed cost (representing the competitor’s prices). The first player chooses an assignment of prices to the blue edges, and the second player then buys the cheapest possible minimum spanning tree, using any combination of red and blue edges. The goal of the first player is to maximize the total price of purchased blue edges. This game is the minimum spanning tree analog of the well-studied Stackelberg shortest-path game. We analyze the complexity and approximability of the first player’s best strategy in StackMST. In particular, we prove that the problem is APX-hard even if there are only two different red costs, and give an approximation algorithm whose approximation ratio is at most min{k, 3 + 2 ln b, 1 + ln W}, where k is the number of distinct red costs, b is the number of blue edges, and W is the maximum ratio between red costs. We also give a natural integer linear programming formulation of the problem, and show that the integrality gap of the fractional relaxation asymptotically matches the approximation guarantee of our algorithm. 1

### Citations

11605 | Johnson: Computers and Intractability: A guide to theory of NP hardness, Feeman and - Garey, S - 1979 |

602 | Combinatorial Optimization — Polyhedra and Efficiency - Schrijver - 2003 |

576 |
Optimization, approximation, and complexity classes
- Papadimitriou, Yannakakis
- 1991
(Show Context)
Citation Context ...ε > 0, a (1 − ε)approximation for StackMST implies a (1 + 8ε)-approximation for VertexCover in graphs of maximum degree at most 3. The theorem then follows from the APX-hardness of the latter problem =-=[1,13]-=-. The formal details will appear in the full version of this paper. 4 The Best-Out-Of-k Algorithm As before, let k denote the number of distinct red costs, and let c1 < c2 < · · · < ck denote those co... |

316 | Approximation algorithms for NP-complete problems on planar graphs - Baker - 1994 |

114 | Stackelberg Scheduling Strategies
- Roughgarden
(Show Context)
Citation Context ...le, Dodis and Roughgarden [4] and Swamy [16]. An extensive bibliography on similar networking games has been compiled recently by Altman et al. [2]. In another Stackelberg game studied by Roughgarden =-=[15]-=-, the leader is a job scheduler whose goal is to compute a scheduling strategy for the jobs he/she controls such that total latency in the system is minimized after the followers have selfishly schedu... |

107 |
On profit-maximizing envy-free pricing
- Guruswami, Hartline, et al.
- 2005
(Show Context)
Citation Context ...e amounts of each item can be sold. APXhardness and approximability of such problems have been established by Hartline and Koltum [13], and by Guruswami, Hartline, Karlin, Kempe, Kenyon, and McSherry =-=[12]-=-. Balcan and Blum [2] gave improved approximation results. Approximability within a logarithmic factor has also been recently established for more general cases by Balcan, Blum and Mansour [3]. The ca... |

100 | Pricing Network Edges for Heterogeneous Selfish Users
- Cole, Dodis, et al.
- 2003
(Show Context)
Citation Context ...mal behaviors among the followers (the users of the network) so as to maximize some global objective. These kinds of Stackelberg games have been studied recently, e.g., by Cole, Dodis and Roughgarden =-=[4]-=- and Swamy [16]. An extensive bibliography on similar networking games has been compiled recently by Altman et al. [2]. In another Stackelberg game studied by Roughgarden [15], the leader is a job sch... |

70 | A survey on networking games in telecommunications
- Altman, Boulogne, et al.
- 2006
(Show Context)
Citation Context ...tackelberg games have been studied recently, e.g., by Cole, Dodis and Roughgarden [4] and Swamy [16]. An extensive bibliography on similar networking games has been compiled recently by Altman et al. =-=[2]-=-. In another Stackelberg game studied by Roughgarden [15], the leader is a job scheduler whose goal is to compute a scheduling strategy for the jobs he/she controls such that total latency in the syst... |

66 | Combination can be hard: Approximability of the unique coverage problem
- Demaine, Feige, et al.
(Show Context)
Citation Context ... [11], and Briest and Krysta [5]. A semi-logarithmic inapproximability result for a special case of the unlimited supply pricing problem has been given by Demaine, Feige, Hajiaghayi, and Salavatipour =-=[9]-=-. Other kinds of Stackelberg games in networks have been studied by Cole, Dodis, and Roughgarden [8], Swamy [19], and Roughgarden [17]. Our results. We analyze the complexity and approximability of th... |

65 | Approximation algorithms and online mechanisms for item pricing
- Balcan, Blum
(Show Context)
Citation Context ...elberg game. APX-hardness and approximability of such problems have been established by Hartline and Koltum [12], and by Guruswami, Hartline, Karlin, Kempe, Kenyon, and McSherry [11]. Balcan and Blum =-=[2]-=- gave improved approximation results. Approximability within a logarithmic factor has also been recently established for more general cases by Balcan, Blum and Mansour [3]. The case in which items are... |

62 |
Some apx-completeness results for cubic graphs
- Alimonti, Kann
(Show Context)
Citation Context ...ε > 0, a (1 − ε)approximation for StackMST implies a (1 + 8ε)-approximation for VertexCover in graphs of maximum degree at most 3. The theorem then follows from the APX-hardness of the latter problem =-=[1,13]-=-. The formal details will appear in the full version of this paper. 4 The Best-Out-Of-k Algorithm As before, let k denote the number of distinct red costs, and let c1 < c2 < · · · < ck denote those co... |

56 | A bilevel model of taxation and its applications to optimal highway pricing. Management Science 44
- Labbé, Marcotte, et al.
- 1998
(Show Context)
Citation Context ... |B|)-approximable. A generalization of the problem to more than one customer has been tackled using mathematical programming tools, in particular bilevel programming; see Labbé, Marcotte, and Savard =-=[12]-=-. This generalization was motivated by the problem of setting tolls on highway networks. Cardinal, Labbé, Langerman, and Palop [3] give a geometric version of the problem. Sometimes the goal of the le... |

48 | Beyond VCG: Frugality of Truthful Mechanisms
- Karlin, Kempe, et al.
- 2005
(Show Context)
Citation Context ...sumer’s behavior in the form of a combinatorial preference structure. Finally, our problem should not be confused with other spanning tree games found in cooperative games and mechanism design theory =-=[8,11]-=-, with parametric spanning tree problems [7,6], or with two-stage stochastic minimum spanning tree problems [5]. Our results. We analyze the complexity and approximability of the StackMST problem. Spe... |

46 |
Minimum cost spanning tree games
- Granot, Huberman
- 1981
(Show Context)
Citation Context ...sumer’s behavior in the form of a combinatorial preference structure. Finally, our problem should not be confused with other spanning tree games found in cooperative games and mechanism design theory =-=[8,11]-=-, with parametric spanning tree problems [7,6], or with two-stage stochastic minimum spanning tree problems [5]. Our results. We analyze the complexity and approximability of the StackMST problem. Spe... |

45 | Single-minded unlimited supply pricing on sparse instances
- Briest, Krysta
(Show Context)
Citation Context ... established for more general cases by Balcan, Blum and Mansour [3]. The case in which items are edges of a graph has been studied by Grigoriev, van Loon, Sitters and Uetz [10], and Briest and Krysta =-=[6]-=-. A semi-logarithmic inapproximability result for a special case of the unlimited supply pricing problem has been given by Demaine, Feige, Hajiaghayi, and Salavatipour [9]. Our results. We analyze the... |

29 | Item pricing for revenue maximization
- Balcan, Blum, et al.
(Show Context)
Citation Context ...Sherry [11]. Balcan and Blum [2] gave improved approximation results. Approximability within a logarithmic factor has also been recently established for more general cases by Balcan, Blum and Mansour =-=[3]-=-. The case in which items are edges of a graph has been studied by Grigoriev, van Loon, Sitters and Uetz [10], and Briest and Krysta [6]. A semi-logarithmic inapproximability result for a special case... |

29 | A polynomial-time approximation scheme for Steiner tree in planar graphs - Borradaile, Kenyon-Mathieu, et al. - 2007 |

28 | The effectiveness of stackelberg strategies and tolls for network congestion games
- Swamy
- 2007
(Show Context)
Citation Context ...among the followers (the users of the network) so as to maximize some global objective. These kinds of Stackelberg games have been studied recently, e.g., by Cole, Dodis and Roughgarden [4] and Swamy =-=[16]-=-. An extensive bibliography on similar networking games has been compiled recently by Altman et al. [2]. In another Stackelberg game studied by Roughgarden [15], the leader is a job scheduler whose go... |

27 | On two-stage stochastic minimum spanning trees
- Dhamdhere, Ravi, et al.
(Show Context)
Citation Context ...with other spanning tree games found in cooperative games and mechanism design theory [8,11], with parametric spanning tree problems [7,6], or with two-stage stochastic minimum spanning tree problems =-=[5]-=-. Our results. We analyze the complexity and approximability of the StackMST problem. Specifically, we prove the following: 1. StackMST is APX-hard, even if there are only two red costs, 1 and 2 (Sect... |

27 | Near-optimal pricing in near-linear time
- Hartline, Koltun
- 2005
(Show Context)
Citation Context ...e goal is to compute a scheduling strategy for the jobs he/she controls such that total latency in the system is minimized after the followers have selfishly scheduled their jobs. Hartline and Koltun =-=[10]-=- propose approximation algorithms for several APXhard pricing problems, where the goal is to find the best prices for a set of items, given knowledge of the consumer’s behavior in the form of a combin... |

14 | Setting parameters by example
- Eppstein
(Show Context)
Citation Context ... preference structure. Finally, our problem should not be confused with other spanning tree games found in cooperative games and mechanism design theory [8,11], with parametric spanning tree problems =-=[7,6]-=-, or with two-stage stochastic minimum spanning tree problems [5]. Our results. We analyze the complexity and approximability of the StackMST problem. Specifically, we prove the following: 1. StackMST... |

14 | How to sell a graph: Guidelines for graph retailers
- Grigoriev, Loon, et al.
(Show Context)
Citation Context ...actor has also been recently established for more general cases by Balcan, Blum and Mansour [3]. The case in which items are edges of a graph has been studied by Grigoriev, van Loon, Sitters and Uetz =-=[11]-=-, and Briest and Krysta [5]. A semi-logarithmic inapproximability result for a special case of the unlimited supply pricing problem has been given by Demaine, Feige, Hajiaghayi, and Salavatipour [9]. ... |

13 | Stackelberg Network Pricing Games
- Briest, Hoefer, et al.
- 2008
(Show Context)
Citation Context ...Langerman, and Palop [8] give a geometric version of the shortest path problem. Recently, part of the results of the current paper have been generalized to other problems by Briest, Hoefer and Krysta =-=[5]-=-. They also exhibit a polynomial-time algorithm for a special case of a Stackelberg vertex cover problem, in which the follower’s problem is to find a minimum vertex cover in a bipartite graph. Other ... |

12 | A subset spanner for planar graphs, with application to subset TSP - Klein - 2006 |

11 | An approximation algorithm for Stackelberg network pricing. Networks 45
- Roch, Savard, et al.
- 2005
(Show Context)
Citation Context ...ces instead of a spanning tree, has been studied 2sin the literature; see van Hoesel [17] for a survey. Complexity and approximability results have recently been obtained by Roch, Savard and Marcotte =-=[14]-=-, and by Grigoriev, van Hoesel, Kraaij, Uetz, and Bouhtou [9]: the problem is strongly NP-hard and O(log |B|)-approximable. A generalization of the problem to more than one customer has been tackled u... |

8 |
Pricing of geometric transportation networks
- Cardinal, Labbé, et al.
- 2005
(Show Context)
Citation Context ..., in particular bilevel programming; see Labbé, Marcotte, and Savard [12]. This generalization was motivated by the problem of setting tolls on highway networks. Cardinal, Labbé, Langerman, and Palop =-=[3]-=- give a geometric version of the problem. Sometimes the goal of the leader is not to invite the followers to use his/her part of the network and maximize his/her own revenue but to encourage socially ... |

8 | Pricing network edges to cross a river
- Grigoriev, Hoesel, et al.
- 2004
(Show Context)
Citation Context ...rature; see van Hoesel [17] for a survey. Complexity and approximability results have recently been obtained by Roch, Savard and Marcotte [14], and by Grigoriev, van Hoesel, Kraaij, Uetz, and Bouhtou =-=[9]-=-: the problem is strongly NP-hard and O(log |B|)-approximable. A generalization of the problem to more than one customer has been tackled using mathematical programming tools, in particular bilevel pr... |

7 | Using sparsification for parametric minimum spanning tree problems
- Fernández-Baca, Slutzki, et al.
- 1996
(Show Context)
Citation Context ... preference structure. Finally, our problem should not be confused with other spanning tree games found in cooperative games and mechanism design theory [8,11], with parametric spanning tree problems =-=[7,6]-=-, or with two-stage stochastic minimum spanning tree problems [5]. Our results. We analyze the complexity and approximability of the StackMST problem. Specifically, we prove the following: 1. StackMST... |

6 |
Marktform und Gleichgewicht (Market Structure and Equilibrium
- Stackelberg
- 1934
(Show Context)
Citation Context ...f the connections that the customer buys from you. This problem can be cast as a Stackelberg game, a type of two-player game named introduced by the German economist Heinrich Freiherr von Stackelberg =-=[18]-=-. In a Stackelberg game, there are two players: the leader moves first, then the follower moves, and then the game is over. The follower thus optimizes its own objective function, knowing the leader’s... |

5 | Computational Aspects of a 2-Player Stackelberg Shortest Paths Tree Game - Bilò, Gualà, et al. - 2008 |

3 | An overview of Stackelberg pricing in networks
- Hoesel
(Show Context)
Citation Context ...s. Related work. A similar pricing problem, where the customer wants to construct a shortest path between two vertices instead of a spanning tree, has been studied 2sin the literature; see van Hoesel =-=[17]-=- for a survey. Complexity and approximability results have recently been obtained by Roch, Savard and Marcotte [14], and by Grigoriev, van Hoesel, Kraaij, Uetz, and Bouhtou [9]: the problem is strongl... |

3 | Pricing bridges to cross a river
- Bouhtou, Grigoriev, et al.
(Show Context)
Citation Context ...esel [17] for a survey. Complexity and approximability results have recently been obtained by Roch, Savard and Marcotte [15], and by Bouhtou, Grigoriev, van Hoesel, van der Kraaij, Spieksma, and Uetz =-=[4]-=-: the problem is strongly NP-hard and O(log |B|)-approximable. A generalization of the problem to more than one customer has been tackled using mathematical programming tools, in particular bilevel pr... |

2 |
volume 24 of Algorithms and Combinatorics
- Polyhedra, B
(Show Context)
Citation Context ...f G according to some ordering e1,e2,... ,eℓ. Then, for every i ∈ {1,2,... ,ℓ}, the number of edges in {e1,e2,...,ei} picked by the algorithm is exactly r({e1,e2,... ,ei}) (see for instance Schrijver =-=[18]-=-). This implies in particular and so we deduce Thus the revenue given by Ti is exactly mi = r(Ri) − r(Ri−1), m ′ i = r(Ri ∪ B) − r(Ri−1 ∪ B), bi = r(Ri−1 ∪ B) − r(Ri−1), mi − m ′ i = bi − bi+1. (5) qi... |

1 | Graph structure and monadic second-order logic: Language theoretical aspects - Springer-Verlag |

1 | DHK09] [DHM07] [GJ79] Erik D. Demaine, MohammadTaghi Hajiaghayi, and Ken-ichi Kawarabayashi. Approximation algorithms via structural results for apex-minor-free graphs - Demaine, Hajiaghayi, et al. - 2008 |

1 | Optimal pricing of capacitated networks - Networks - 2009 |