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

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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.

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Notes

  1. We suppose that every vertex has at most one degree-1 neighbor. If not, then we can discard some degree-1 vertices of the graph without modifying the size of a maximum matching. In addition, it is not difficult to see that mfv remains the same on the reduced instance.

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Acknowledgments

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

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Authors and Affiliations

  1. Sorbonne Universités, UPMC Université Paris 06, CNRS, LIP6 UMR 7606, 4 place Jussieu, 75005, Paris, France

    Bruno Escoffier

  2. CNRS, UMR 7243, 75775, Paris, France

    Laurent Gourvès & Jérôme Monnot

  3. Lamsade, Université de Paris-Dauphine, 75775, Paris, France

    Laurent Gourvès & Jérôme Monnot

Authors
  1. Bruno Escoffier
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  2. Laurent Gourvès
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  3. Jérôme Monnot
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Corresponding author

Correspondence to Laurent Gourvès.

Additional information

Part of the results given in this article were presented at the conference SAGT 2011 (B. Escoffier, L. Gourvès, J. Monnot: The Price of Optimum in a Matching Game. SAGT 2011: 81–92).

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Escoffier, B., Gourvès, L. & Monnot, J. The Price of Optimum: Complexity and Approximation for a Matching Game. Algorithmica 77, 836–866 (2017). https://doi.org/10.1007/s00453-015-0108-5

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  • DOI: https://doi.org/10.1007/s00453-015-0108-5

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