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Algorithmica

, Volume 77, Issue 3, pp 836–866 | Cite as

The Price of Optimum: Complexity and Approximation for a Matching Game

  • Bruno Escoffier
  • Laurent GourvèsEmail author
  • Jérôme Monnot
Article

Abstract

This paper deals with a matching game in which the nodes of a simple graph are independent agents who try to form pairs. If we let the agents make their decision without any central control then a possible outcome is a Nash equilibrium, that is a situation in which no unmatched player can change his strategy and find a partner. However, there can be a big difference between two possible outcomes of the same instance, in terms of number of matched nodes. A possible solution is to force all the nodes to follow a centrally computed maximum matching but it can be difficult to implement this approach. This article proposes a tradeoff between the total absence and the full presence of a central control. Concretely, we study the optimization problem where the action of a minimum number of agents is centrally fixed and any possible equilibrium of the modified game must be a maximum matching. In algorithmic game theory, this approach is known as the price of optimum of a game. For the price of optimum of the matching game, deciding whether a solution is feasible is not straightforward, but we prove that it can be done in polynomial time. In addition, the problem is shown APX-hard, since its restriction to graphs admitting a perfect matching is equivalent, from the approximability point of view, to vertex cover. Finally we prove that this problem admits a polynomial 6-approximation algorithm in general graphs.

Keywords

Approximation algorithm Srategic game Complexity Price of Anarchy Price of optimum Stackelberg strategy 

Notes

Acknowledgments

We would like to thank the anonymous reviewers for their valuable comments and suggestions.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Bruno Escoffier
    • 1
  • Laurent Gourvès
    • 2
    • 3
    Email author
  • Jérôme Monnot
    • 2
    • 3
  1. 1.Sorbonne Universités, UPMC Université Paris 06, CNRS, LIP6 UMR 7606ParisFrance
  2. 2.CNRS, UMR 7243ParisFrance
  3. 3.LamsadeUniversité de Paris-DauphineParisFrance

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