Skip to main content
SpringerLink
Log in
Book cover

International Conference on Algorithms and Complexity

CIAC 2017: Algorithms and Complexity pp 310–321Cite as

Approximate Maximin Share Allocations in Matroids

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10236))

Abstract

The maximin share guarantee is, in the context of allocating indivisible goods to a set of agents, a recent fairness criterion. A solution achieving a constant approximation of this guarantee always exists and can be computed in polynomial time. We extend the problem to the case where the goods collectively received by the agents satisfy a matroidal constraint. Polynomial approximation algorithms for this generalization are provided: a 1/2-approximation for any number of agents, a \((1-\varepsilon )\)-approximation for two agents, and a \((8/9-\varepsilon )\)-approximation for three agents. Apart from the extension to matroids, the \((8/9-\varepsilon )\)-approximation for three agents improves on a \((7/8-\varepsilon )\)-approximation by Amanatidis et al. (ICALP 2015).

Supported by the project ANR CoCoRICo-CoDec.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Brams, S.J., Taylor, A.D.: Fair Division: From Cake Cutting to Dispute Resolution. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  2. Bouveret, S., Lemaître, M.: Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Auton. Agents Multi-agent Syst. 30, 259–290 (2016)

    Article  Google Scholar 

  3. Budish, E.: The combinatorial assignment problem: approximate competitive equilibrium from equal incomes. J. Polit. Econ. 119, 1061–1103 (2011)

    Article  Google Scholar 

  4. Procaccia, A.D., Wang, J.: Fair enough: guaranteeing approximate maximin shares. In: Babaioff, M., Conitzer, V., Easley, D. (eds.) ACM Conference on Economics and Computation, EC 2014, Stanford, CA, USA, 8–12 June 2014, pp. 675–692. ACM (2014)

    Google Scholar 

  5. Kurokawa, D., Procaccia, A.D., Wang, J.: When can the maximin share guarantee be guaranteed? In: Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, 12–17 February 2016, Phoenix, Arizona, USA, pp. 523–529. AAAI Press (2016)

    Google Scholar 

  6. Lipton, R., Markakis, E., Mossel, E., Saberi, A.: On approximately fair allocations of indivisible goods. In: Breese, J., Feigenbaum, J., Seltzer, M. (eds.) ACM Conference on Electronic Commerce, pp. 125–131. ACM (2004)

    Google Scholar 

  7. Asadpour, A., Saberi, A.: An approximation algorithm for max-min fair allocation of indivisible goods. SIAM J. Comput. 39, 2970–2989 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Markakis, E., Psomas, C.-A.: On worst-case allocations in the presence of indivisible goods. In: Chen, N., Elkind, E., Koutsoupias, E. (eds.) WINE 2011. LNCS, vol. 7090, pp. 278–289. Springer, Heidelberg (2011). doi:10.1007/978-3-642-25510-6_24

    Chapter  Google Scholar 

  9. Amanatidis, G., Markakis, E., Nikzad, A., Saberi, A.: Approximation algorithms for computing maximin share allocations. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 39–51. Springer, Heidelberg (2015). doi:10.1007/978-3-662-47672-7_4

    Google Scholar 

  10. Woeginger, G.J.: A polynomial-time approximation scheme for maximizing the minimum machine completion time. Oper. Res. Lett. 20, 149–154 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Spliddit: provably fair solutions (2017). http://www.spliddit.org/

  12. Brualdi, R.: Comments on bases in different structures. Bull. Austral. Math. Soc. 1, 161–167 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  13. Greene, C.: A multiple exchange property for bases. Proc. Am. Math. Soc. 39, 45–50 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  14. Woodall, D.: An exchange theorem for bases of matroids. J. Comb. Theory (B) 16, 227–228 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  15. Greene, C., Magnanti, T.L.: Some abstract pivot algorithms. SIAM J. Appl. Math. 29, 530–539 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bixby, R.E., Cunningham, W.H.: Matroid optimization and algorithms. In: Handbook of Combinatorics, North Holland (1995)

    Google Scholar 

  17. Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 4th edn. Springer Publishing Company, Incorporated, Heidelberg (2007)

    MATH  Google Scholar 

  18. Gourvès, L., Monnot, J., Tlilane, L.: A protocol for cutting matroids like cakes. In: Chen, Y., Immorlica, N. (eds.) WINE 2013. LNCS, vol. 8289, pp. 216–229. Springer, Heidelberg (2013). doi:10.1007/978-3-642-45046-4_18

    Chapter  Google Scholar 

  19. Korte, B., Lovász, L., Schrader, R.: Greedoids. Springer, Heidelberg (1991)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, 75016, Paris, France

    Laurent Gourvès & Jérôme Monnot

Authors
  1. Laurent Gourvès
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jérôme Monnot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laurent Gourvès .

Editor information

Editors and Affiliations

  1. National Technical University of Athens, Zografou, Athens, Greece

    Dimitris Fotakis

  2. National Technical University of Athens, Zografou, Athens, Greece

    Aris Pagourtzis

  3. Université Paris-Dauphine, Paris, France

    Vangelis Th. Paschos

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Gourvès, L., Monnot, J. (2017). Approximate Maximin Share Allocations in Matroids. In: Fotakis, D., Pagourtzis, A., Paschos, V. (eds) Algorithms and Complexity. CIAC 2017. Lecture Notes in Computer Science(), vol 10236. Springer, Cham. https://doi.org/10.1007/978-3-319-57586-5_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-57586-5_26

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57585-8

  • Online ISBN: 978-3-319-57586-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation