Resource Competition on Integral Polymatroids

Harks, Tobias; Klimm, Max; Peis, Britta

doi:10.1007/978-3-319-13129-0_14

Cited by 16 publications

(37 citation statements)

References 32 publications

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“…We assume that g i is given through a value oracle. Polymatroid congestion games are a generalization of independent set matroid congestion games where the strategies of each player are the integer vectors in an integer polymatroid, see [10]. In base polymatroid congestion games, for each i = 1, .…”

Section: Matroid and Polymatroid Congestion Gamesmentioning

confidence: 99%

Totally Unimodular Congestion Games

Pia¹,

Ferris²,

Michini³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We investigate a new class of congestion games, called Totally Unimodular (TU) Congestion Games, where the players' strategies are binary vectors inside polyhedra defined by totally unimodular constraint matrices. Network congestion games belong to this class.In the symmetric case, when all players have the same strategy set, we design an algorithm that finds an optimal aggregated strategy and then decomposes it into the single players' strategies. This approach yields strongly polynomial-time algorithms to (i) find a pure Nash equilibrium, and (ii) compute a socially optimal state, if the delay functions are weakly convex. We also show how this technique can be extended to matroid congestion games.We then introduce some combinatorial TU congestion games, where the players' strategies are matchings, vertex covers, edge covers, and stable sets of a given bipartite graph. In the asymmetric case, we show that for these games (i) it is PLS-complete to find a pure Nash equilibrium even in case of linear delay functions, and (ii) it is NP-hard to compute a socially optimal state, even in case of weakly convex delay functions.

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Section: Matroid and Polymatroid Congestion Gamesmentioning

confidence: 99%

Totally Unimodular Congestion Games

Pia¹,

Ferris²,

Michini³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In contrast to our model, the models above allowed that a player uses a resource with multiple units of demand at the same time. It turns out that allowing for this kind of self-congestion has a severe impact on the existence of pure Nash equilibria [10,24] but for networks of parallel links it is known that pure Nash equilibria are guranteed to exist [16,28].…”

Section: Related Workmentioning

confidence: 99%

Efficiency of Equilibria in Uniform Matroid Congestion Games

Jong

Klimm

Uetz

2016

Algorithmic Game Theory

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Network routing games, and more generally congestion games play a central role in algorithmic game theory, comparable to the role of the traveling salesman problem in combinatorial optimization. It is known that the price of anarchy is independent of the network topology for non-atomic congestion games. In other words, it is independent of the structure of the strategy spaces of the players, and for a ne cost functions it equals 4/3. In this paper, we show that the dependence of the price of anarchy on the network topology is considerably more intricate for atomic congestion games. More specifically, we consider congestion games with a ne cost functions where the strategy spaces of players are symmetric and equal to the set of bases of a k-uniform matroid. In this setting, we show that the price of anarchy is strictly larger than the price of anarchy for singleton strategy spaces where the latter is 4/3. As our main result we show that the price of anarchy can be bounded from above by 28/13 ¥ 2.15. This constitutes a substantial improvement over the price of anarchy bound 5/2, which is known to be tight for arbitrary network routing games with a ne cost functions. ACM Subject Classification IntroductionUnderstanding the impact of selfish behavior on the performance of a system is an important question in algorithmic game theory. One of the cornerstones of the substantial literature on this topic is the famous result of Roughgarden and Tardos [26]. They considered the tra c model of Wardrop [29] in a network with a ne flow-dependent congestion cost functions on the edges. Given a set of commodities, each specified by a source node, a target node, and a flow demand, a Wardrop equilibrium is a multicommodity flow with the property that every commodity uses only paths that minimize the cost. For this setting, Roughgarden and Tardos proved that the total cost of an equilibrium flow is not worse than 4/3 times that of a system optimum. This ratio was coined the price of anarchy by Koutsoupias and Papadimitriou [18] who introduced it as a measure of a system's performance degradation due to selfish behavior. A surprising consequence of the result of Roughgarden and Tardos is that the worst case price of anarchy in congested networks is attained for very simple single-commodity networks already considered a century ago by Pigou [22]. Pigou-style networks consist of only two nodes connected by two parallel links. In fact, Roughgarden [25] proved that for any set of cost functions, the price of anarchy is independent of the network topology as it is always

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“…We remark that polymatroid congestion games were recently introduced by Harks et al [14]. In contrast to their model, we do not allow that cost functions are player-specific, but we do allow general non-decreasing cost functions instead of convex cost functions.…”

Section: Polymatroid Congestion Modelsmentioning

confidence: 99%

A logarithmic approximation for polymatroid congestion games

Harks

Oosterwijk

Vredeveld

2016

Operations Research Letters

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We study the problem of computing a social optimum (minimum cost solution) in polymatroid congestion games, where the strategy space of every player consists of the set of vectors in a playerspecific integral polymatroid base polyhedron defined on the ground set of resources. For general non-decreasing cost functions we devise an H rk -approximation algorithm, where rk is the sum of the ranks of the player-specific polymatroids and H rk denotes the rk-th harmonic number. The main idea of our algorithm is to iteratively increase resource utilization in a greedy fashion and to invoke a polynomial covering oracle that checks feasibility of every computed resource utilization. The approximation guarantee is best possible up to a constant factor. As a special case, our result 2) where the complexity of computing a socially optimal solution for matroid congestion games with non-decreasing cost functions is considered. Here, the approximation guarantee is best possible up to a constant factor if the number of resources is polynomially bounded in the number of players.

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Resource Competition on Integral Polymatroids

Cited by 16 publications

References 32 publications

Totally Unimodular Congestion Games

Totally Unimodular Congestion Games

Efficiency of Equilibria in Uniform Matroid Congestion Games

A logarithmic approximation for polymatroid congestion games

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