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Chore division on a graph

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Abstract

The paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent’s share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies.

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Notes

  1. Consider all the possible divisions of the cake into n disjoint pieces. The maximim share of an agent is her utility for the worst piece taken in an allocation that is most favourable for her in this respect. An MMS allocation is one where each agent is ensured a piece with utility at least her maximin share.

  2. In this and the following example the disutilities in the CCD instances (tables in the left) are normalized to 10 and the utilities in the corresponding dual CFD instances to 30 (tables in the right).

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Acknowledgements

This work has been supported by the bilateral Slovak-French Grant of Campus France PHC STEFANIK 2018, 40562NF and Slovak Research and Development Agency APVV SK-FR-2017-0022. Sylvain Bouveret is also supported by the Project ANR-14-CE24-0007-01 CoCoRICo-CoDec. Katarína Cechlárová is also supported by VEGA Grants 1/0311/18 and 1/0056/18. Julien Lesca is also supported by the CNRS PEPS project JCJC Mappoleon. The authors would also like to thank anonymous referees who helped to improve the paper.

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Authors and Affiliations

  1. CNRS, Grenoble INP, LIG, Univ. Grenoble Alpes, 38000, Grenoble, France

    Sylvain Bouveret

  2. P.J. Šafárik University, Kosice, Slovakia

    Katarína Cechlárová

  3. PSL, CNRS, LAMSADE, Université Paris-Dauphine, Paris, France

    Julien Lesca

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  1. Sylvain Bouveret
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Correspondence to Sylvain Bouveret.

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Bouveret, S., Cechlárová, K. & Lesca, J. Chore division on a graph. Auton Agent Multi-Agent Syst 33, 540–563 (2019). https://doi.org/10.1007/s10458-019-09415-z

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