## On the impact of combinatorial structure on congestion games

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Venue: | FOCS |

Citations: | 46 - 12 self |

### BibTeX

@ARTICLE{Ackermann_onthe,

author = {Heiner Ackermann and Heiko Röglin and Berthold Vöcking},

title = {On the impact of combinatorial structure on congestion games},

journal = {FOCS},

year = {},

pages = {2006}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the impact of combinatorial structure in congestion games on the complexity of computing pure Nash equilibria and the convergence time of best response sequences. In particular, we investigate which properties of the strategy spaces of individual players ensure a polynomial convergence time. We show that if the strategy space of each player consists of the bases of a matroid over the set of resources, then the lengths of all best response sequences are polynomially bounded in the number of players and resources. We also prove that this result is tight, that is, the matroid property is a necessary and sufficient condition on the players ’ strategy spaces for guaranteeing polynomial time convergence to a Nash equilibrium. In addition, we present an approach that enables us to devise hardness proofs for various kinds of combinatorial games, including first results about the hardness of market sharing games and congestion games for overlay network design. Our approach also yields a short proof for the PLS-completeness of network congestion games. In particular, we show that network congestion games are PLS-complete for directed and undirected networks even in case of linear latency functions.

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Citation Context ...es In this section, we consider matroid congestion games. Before we give a formal definition of such games, we briefly introduce matroids. For a detailed discussion of matroids we refer the reader to =-=[14]-=-. Definition 2.1. A tuple M = (R, I) is a matroid if R is a finite set of resources and I is a nonempty family of subsets of R such that if I 2 I and J ` I, then J 2 I, and if I, J 2 I and |J | < |I|,... |

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Citation Context ...ciated with her selected resources. Almost needless to say, congestion games are fundamental to routing, network design, and other kinds of resource sharing problems in distributed systems. Rosenthal =-=[12]-=- shows with a potential function argument that every congestion game possesses at least one pure Nash equilibrium. This argument does not only prove the existence of pure Nash equilibria but it also s... |

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Citation Context ...n embedding reduction. 4.4 Overlay Network Design Games An overlay network is a network built on top of another network with fixed routing paths between all pairs of nodes. For example, Stoica et al. =-=[15]-=- suggest to generalize the Internet point to point communication to provide services like multicast, anycast, and mobility on the basis of overlay networks. In the case of multicast and anycast the ov... |

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Citation Context ...nique for constructing instances of local search problems with exponentially long sequences of local improvements is to construct instances that resemble the behavior of a binary counter (see, e. g., =-=[2, 6, 10]-=-). We construct a game that consists of n gadgets G0, . . . , Gn-1 that correspond to the bits of the counter. Each of these gadgets has a 0-state and a 1-state and for each gadget there exists a best... |

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Citation Context ...ilibrium of the symmetric game can be transformed into a Nash equilibrium of the original game in polynomial time. 4.3 Market Sharing Games Market Sharing games have been introduced by Goemans et al. =-=[5]-=- to model non-cooperative content distribution in wireless networks. An instance of a market sharing game consists of a set N = {1, . . . , n} of players, a set M with |M| = l of markets, and a bipart... |

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Citation Context ...ly prove the existence of pure Nash equilibria but it also shows that such an equilibrium is reached in a natural way when players iteratively play best responses. A recent result of Fabrikant et al. =-=[4]-=- shows, however, that these best response sequences may require an exponential number of iterations. Their analysis relates congestion games to local search problems and it shows that it is PLS-comple... |

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Citation Context ... best response requires to solve a shortest path problem. On the other extreme, we find singleton games in which all of the players' strategies consist only of single resources. Recently Ieong et al. =-=[7]-=- have shown that best response sequences for singleton games reach a Nash equilibrium after only a polynomial number of iterations. This result can be seen as a criterion on the strategy space of the ... |

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Citation Context ...nique for constructing instances of local search problems with exponentially long sequences of local improvements is to construct instances that resemble the behavior of a binary counter (see, e. g., =-=[2, 6, 10]-=-). We construct a game that consists of n gadgets G0, . . . , Gn-1 that correspond to the bits of the counter. Each of these gadgets has a 0-state and a 1-state and for each gadget there exists a best... |

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Citation Context ...nimizing the social delay fl(S) = Pi2N ffii(S) among all Nash equilibria, or a socially optimal state, i. e., a state S minimizing the social delay among all states. Chakrabarty, Mehta, and Nagarajan =-=[3]-=- present an efficient algorithm for computing a socially optimal state of a singleton congestion game with monotone delay functions. Ieong et al. [7] present an efficient algorithm for computing a soc... |

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Citation Context ...nique for constructing instances of local search problems with exponentially long sequences of local improvements is to construct instances that resemble the behavior of a binary counter (see, e. g., =-=[2, 6, 10]-=-). We construct a game that consists of n gadgets G0, . . . , Gn-1 that correspond to the bits of the counter. Each of these gadgets has a 0-state and a 1-state and for each gadget there exists a best... |

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Citation Context ... combinatorial structures when various players with possibly different strategy spaces are involved. Illustrative examples based on uniform matroids are market sharing games with uniform market costs =-=[5, 9]-=-, and scheduling games in which each player has to injectively allocate a given set of tasks (services) to a given set of machines (servers). We 5salso analyze the complexity of finding socially optim... |

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Citation Context ...optimum of instance f(I), then g(S 2 , I) is a local optimum of I. Additionally, a local search problem Π in PLS is PLS-complete if every problem in PLS is PLS-reducible to Π. Schäffer and Yannakakis =-=[13]-=- introduce the notion of a tight PLS-reduction, which ensures several properties of the corresponding transition graphs. Definition 1.3. A PLS-reduction (f, g) is tight if for any instance I ∈ IΠ1 one... |

2 |
and Mihalis Yannakakis. Simple local search problems that are hard to solve
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Citation Context ...um of instance f (I), then g(S2, I) is a local optimum of I. Additionally, a local search problem \Pisin PLS is PLS-complete if every problem in PLS is PLS-reducible to \Pi . Sch"affer and Yannakakis =-=[13]-=- introduce the notion of a tight PLS-reduction, which ensures several properties of the corresponding transition graphs. Definition 1.3. A PLS-reduction (f, g) is tight if for any instance I 2 I\Pi 1 ... |

2 |
João Setubal, and Arlindo da Conceicão. Finding minimum congestion spanning trees
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Citation Context ...elay by at least a factor of " > 1. The obvious application of matroid congestion games are network design problems in which players compete for the edges of a graph in order to build a spanning tree =-=[16]-=-. There are quite a few more interesting applications as even simple matroid structures like uniform matroids, which are rather uninteresting from an optimization point of view, lead to rich combinato... |

1 | compact: A simple class of congestion games - Fast |