Congestion games viewed from M-convexity

Fujishige, Satoru; Goemans, Michel X.; Harks, Tobias; Peis, Britta; Zenklusen, Rico

doi:10.1016/j.orl.2015.04.002

Cited by 10 publications

(8 citation statements)

References 24 publications

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“…In a nutshell, Theorem 1.2 follows from the fact that in EP congestion games Rosenthal's potential is M-convex, as was shown by Fujishige et al [22]. M-convexity is a property defined in the area of discrete convex analysis [41] (see Section 2.2).…”

Section: Our Contributions and Techniquesmentioning

confidence: 83%

“…We first address the questions from the introduction for extension parallel (EP) congestion games, a well-studied special case of symmetric network congestion games, see, e.g., [28,21,22]. Here, the common strategy set of all players is given by the set of o, d-paths P of an extension parallel graph (see Section 4 for a definition and example).…”

Section: Our Contributions and Techniquesmentioning

confidence: 99%

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Sampling from the Gibbs Distribution in Congestion Games

Kleer¹

2021

Preprint

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Logit dynamics is a form of randomized game dynamics where players have a bias towards strategic deviations that give a higher improvement in cost. It is used extensively in practice. In congestion (or potential) games, the dynamics converges to the so-called Gibbs distribution over the set of all strategy profiles, when interpreted as a Markov chain. In general, logit dynamics might converge slowly to the Gibbs distribution, but beyond that, not much is known about their algorithmic aspects, nor that of the Gibbs distribution. In this work, we are interested in the following two questions for congestion games: i) Is there an efficient algorithm for sampling from the Gibbs distribution? ii) If yes, do there also exist natural randomized dynamics that converges quickly to the Gibbs distribution?We first study these questions in extension parallel congestion games, a well-studied special case of symmetric network congestion games. As our main result, we show that there is a simple variation on the logit dynamics (in which we in addition are also allowed to randomly interchange the strategies of two players) that converges quickly to the Gibbs distribution in such games. This answers both questions above affirmatively. We also address the first question for the class of so-called capacitated k-uniform congestion games.To prove our results, we rely on the recent breakthrough work of Anari, Liu, Oveis-Gharan and Vinzant (2019) concerning the approximate sampling of the base of a matroid according to strongly log-concave probability distribution.

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Section: Our Contributions and Techniquesmentioning

confidence: 83%

Section: Our Contributions and Techniquesmentioning

confidence: 99%

Sampling from the Gibbs Distribution in Congestion Games

Kleer¹

2021

Preprint

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“…Interestingly, by observing the close relationship between this potential function Φ( P ) and M‐convexity, Fujishige et al. (2015) point out that the above best‐response algorithm can be derived from the steepest descent algorithm for M‐convex function minimization problems mentioned in section 3. To see the relationship between M‐convexity and Φ, denote by

Q_{a}

the set of st ‐paths containing arc a .…”

Section: Applicationsmentioning

confidence: 99%

Discrete Convex Analysis and Its Applications in Operations: A Survey

Chen

2021

Production and Operations Management

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D iscrete convexity, in particular, L \ -convexity and M \ -convexity, provides a critical opening to attack several classical problems in inventory theory, as well as many other operations problems that arise from more recent practices, for instance, appointment scheduling and bike sharing. As a powerful framework, discrete convex analysis is becoming increasingly popular in the literature. This review will survey the landscape of the approach. We start by introducing several key concepts, namely, L \ -convexity and M \ -convexity and their variants, followed by a discussion of some fundamental properties that are most useful for studying operations models. We then illustrate various applications of these concepts and properties. Examples include network flow problem, stochastic inventory control, appointment scheduling, game theory, portfolio contract, discrete choice model, and bike sharing. We focus our discussion on demonstrating how discrete convex analysis can shed new insights on existing problems, and/or bring about much more simpler analyses and algorithm developments than previous methods in the literature. We also present several results and analyses that are new to the literature.

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“…Fujishige et al [27] draw an interesting connection between the convergence of best response dynamics and Rosenthal's potential function. In particular, they show that the fast convergence of best response dynamics for congestion games on extensionparallel networks shown by Fotakis [24] follows from the M-convexity of the potential function in this case.…”

Section: Further Comparison With Related Workmentioning

confidence: 99%

Computation and efficiency of potential function minimizers of combinatorial congestion games

Kleer

Schäfer

2020

Math. Program.

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We study the computation and efficiency of pure Nash equilibria in combinatorial congestion games, where the strategies of each player i are given by the binary vectors of a polytope P i. Our main goal is to understand which structural properties of such polytopal congestion games enable us to derive an efficient equilibrium selection procedure to compute pure Nash equilibria with attractive social cost approximation guarantees. To this aim, we identify two general properties of the underlying aggregation polytope P N = i P i which are sufficient for our results to go through, namely the integer decomposition property (IDP) and the box-totally dual integrality property (box-TDI). Our main results for polytopal congestion games satisfying IDP and box-TDI are as follows: (i) we show that pure Nash equilibria can be computed in polynomial time. In fact, we obtain this result through a general framework for separable convex function minimization, which might be of independent interest. (ii) We bound the inefficiency of these equilibria and show that this provides a tight bound on the price of stability. (iii) We also prove that these results extend to strong equilibria for the "bottleneck variant" of polytopal congestion games. Examples of polytopal congestion games satisfying IDP and box-TDI include common source network congestion games, symmetric totally unimodular congestion games, non-symmetric matroid congestion games and symmetric matroid intersection congestion games (in particular, r-arborescences and strongly base-orderable matroids).

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Congestion games viewed from M-convexity

Cited by 10 publications

References 24 publications

Sampling from the Gibbs Distribution in Congestion Games

Sampling from the Gibbs Distribution in Congestion Games

Discrete Convex Analysis and Its Applications in Operations: A Survey

Computation and efficiency of potential function minimizers of combinatorial congestion games

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