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Computing the nucleolus of weighted cooperative matching games in polynomial time

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Abstract

We provide an efficient algorithm for computing the nucleolus for an instance of a weighted cooperative matching game. This resolves a long-standing open question posed in Faigle (Math Programm, 83: 555–569, 1998).

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Notes

  1. It is common within the literature, for instance in [30], to exclude the coalitions for \(S = \varnothing \) and \(S = V\) in the definition of the nucleolus. On the other hand, one could also consider the definition of the nucleolus with all possible coalitions, including \(S = \varnothing \) and \(S = V\). We note that the two definitions of the nucleolus are equivalent in all instances of matching games except for the trivial instance of a graph consisting of two nodes joined by a single edge.

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Acknowledgements

The authors thank Umang Bhaskar, Daniel Dadush, and Linda Farczadi for stimulating and insightful discussions related to this paper.

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Authors and Affiliations

  1. University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Jochen Könemann & Justin Toth

  2. University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    Kanstantsin Pashkovich

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  1. Jochen Könemann
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  2. Kanstantsin Pashkovich
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Correspondence to Justin Toth.

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This work was done in part while the second author was visiting the Simons Institute for the Theory of Computing. Supported by DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant #CCF-1740425.

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG).

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Könemann, J., Pashkovich, K. & Toth, J. Computing the nucleolus of weighted cooperative matching games in polynomial time. Math. Program. 183, 555–581 (2020). https://doi.org/10.1007/s10107-020-01483-4

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