The Fair Division of Hereditary Set Systems

  • Z. LiEmail author
  • A. Vetta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)


We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form an hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least 0.3667 times the maximin share of the agent. This improves upon the current best known guarantee of 0.2 due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most 0.3738. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.



The authors thank Jugal Garg, Vasilis Gkatzelis and Richard Santiago for interesting discussions on fair division. We thank the anonymous reviewers for helpful suggestions.


  1. 1.
    Amanatidis, G., Markakis, E., Nikzad, A., Saberi, A.: Approximation algorithms for computing maximin share allocations. ACM Trans. Algorithms 13(4) (2018). Article #52Google Scholar
  2. 2.
    Barman, S., Krishna Murthy, S.: Approximation algorithms for maximin fair division. In: Proceedings of the 18th Conference on Economics and Computation (EC), pp. 647–664 (2017)Google Scholar
  3. 3.
    Bouveret, S., Cechlarova, K., Elkind, E., Igarashi, A., Peters, D.: Fair division of a graph. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI), pp. 135–141 (2017)Google Scholar
  4. 4.
    Bouveret, S., Lang, J.: Characterizing conflicts in fair division of indivisible goods using a scale of criteria. In: Proceedings of 13th Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pp. 1321–1328 (2014)Google Scholar
  5. 5.
    Brahms, S., King, D.: Fair Division: From Cake Cutting to Dispute Resolution. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  6. 6.
    Budish, E.: The combinatorial assignment problem: approximate competitive equilibrium from equal incomes. J. Polit. Econ. 119(6), 1061–1103 (2011)CrossRefGoogle Scholar
  7. 7.
    Caragiannis, I., Kurokawa, D., Moulin, H., Procaccia, A., Shah, N., Wang, J.: The unreasonable fairness of maximum nash welfare. In: Proceedings of 16th Conference on Economics and Computation (EC), pp. 305–322 (2016)Google Scholar
  8. 8.
    Ghodsi, M., HajiAghayi, M., Seddighin, M., Seddighin, S., Yami, H.: Fair allocation of indivisible goods: Improvement and generalization. In: Proceedings of the 19th Conference on Economics and Computation (EC), pp. 539–556 (2018)Google Scholar
  9. 9.
    Kurokawa, D., Procaccia, A., Wang, J.: Fair enough: guaranteeing approximate maximin shares. J. ACM 65(2) (2018). Article #8CrossRefGoogle Scholar
  10. 10.
    Moulin, H.: Uniform externalities: two axioms for fair allocation. J. Public Econ. 43(3), 305–326 (1990)CrossRefGoogle Scholar
  11. 11.
    Moulin, H.: Fair Division and Collective Welfare. MIT Press, Cambridge (2003)Google Scholar
  12. 12.
    Robertson, J., Webb, W.: Cake Cutting Algorithms: Be Fair If You Can. A K Peters, Natick (1998)CrossRefGoogle Scholar
  13. 13.
    Steinhaus, H.: The problem of fair division. Econometrica 16, 101–104 (1948)Google Scholar
  14. 14.
    Varian, H.: Equity, envy and efficiency. J. Econ. Theory 9, 63–91 (1974)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Woeginger, G.: A polynomial time approximation scheme for maximizing the minimum machine completion time. Oper. Res. Lett. 20, 149–154 (1997)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Computer Science DepartmentÉcole Normale SupérieureParisFrance
  2. 2.Department of Mathematics and Statistics, and School of Computer ScienceMcGill UniversityMontrealCanada

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