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Agreeable sets with matroidal constraints

  • Laurent Gourvès
Article
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Abstract

This article deals with the challenge of reaching an agreement for a group of agents who have heterogeneous preferences over a set of goods. In a recent work, Suksompong (in: Subbarao (ed) Proceedings of the twenty-fifth international joint conference on artificial intelligence, IJCAI 2016, New York, pp 489–495, 2016) models a problem of this kind as the search of an agreeable subset of a given ground set of goods. A subset is agreeable if it is weakly preferred to its complement by every agent of the group. Under natural assumptions on the agents’ preferences such as monotonicity or responsiveness, an agreeable set of small cardinality is guaranteed to exist, and it can be efficiently computed. This article deals with an extension to subsets which must satisfy extra matroidal constraints. Worst case upper bounds on the size of an agreeable set are shown, and algorithms for computing them are given. For the case of two agents having additive preferences, we show that an agreeable solution can also be approximately optimal (up to a multiplicative constant factor) for both agents.

Keywords

Allocation of indivisible goods Matroids Approximation 

References

  1. Barberà S, Bossert W, Pattanaik PK (2004) Ranking sets of objects. In: Barberà S, Hammond P, Seidl C (eds) Handbook of utility theory, chapter 17. Kluwer Academic Publishers, BostonCrossRefGoogle Scholar
  2. Biró P, Fleiner T, Irving RW, Manlove D (2010) The college admissions problem with lower and common quotas. Theor Comput Sci 411(34–36):3136–3153MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bouveret S, Chevaleyre Y, Maudet N (2016) Fair allocation of indivisible goods. In: Brandt et al. (eds) Handbook of computational social choice. Cambridge University Press, Cambridge, pp 284–310Google Scholar
  4. Brandt F, Conitzer V, Endriss U, Lang J, Procaccia AD (eds) (2016) Handbook of computational social choice. Cambridge University Press, CambridgeGoogle Scholar
  5. Ferraioli D, Gourvès L, Monnot J (2014) On regular and approximately fair allocations of indivisible goods. In: Bazzan ALC, Huhns MN, Lomuscio A, Scerri P (eds) International conference on autonomous agents and multi-agent systems, AAMAS ’14, Paris, France, May 5–9, 2014, pp 997–1004. IFAAMAS/ACMGoogle Scholar
  6. Foley DK (1967) Resource allocation and the public sector. Yale Econ Essays 7(1):45–98Google Scholar
  7. Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69(1):9–15MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gourvès L, Martinhon CA, Monnot J (2016) Object allocation problems under constraints. In: COMSOCGoogle Scholar
  9. Gourvès L, Monnot J, Tlilane L (2013) A matroid approach to the worst case allocation of indivisible goods. In: Rossi F (ed) IJCAI 2013, proceedings of the 23rd international joint conference on artificial intelligence, Beijing, China, August 3–9, 2013, pp 136–142. IJCAI/AAAIGoogle Scholar
  10. Gourvès L, Monnot J, Tlilane L (2015) Approximate tradeoffs on weighted labeled matroids. Discrete Appl Math 184:154–166MathSciNetCrossRefzbMATHGoogle Scholar
  11. Gusfield D, Irving RW (1989) The stable marriage problem: structure and algorithms. MIT Press, CambridgezbMATHGoogle Scholar
  12. Korte B, Vygen J (2007) Combinatorial optimization: theory and algorithms, 4th edn. Springer, BerlinzbMATHGoogle Scholar
  13. Manurangsi P, Suksompong W (2017) Computing an approximately optimal agreeable set of items. In: Sierra Ca (ed) Proceedings of the twenty-sixth international joint conference on artificial intelligence, IJCAI 2017, Melbourne, Australia, August 19–25, 2017, pp 338–344. www.ijcai.org. Accessed 17 Apr 2018
  14. Oxley JG (1992) Matroid theory. Oxford University Press, OxfordzbMATHGoogle Scholar
  15. Rothe J (ed) (2016) Economics and computation, an introduction to algorithmic game theory, computational social choice, and fair division. Springer, BerlinzbMATHGoogle Scholar
  16. Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. Springer, BerlinzbMATHGoogle Scholar
  17. Steven J, Marc Kilgour D, Klamler C (2012) The undercut procedure: an algorithm for the envy-free division of indivisible items. Soc Choice Welf 39(2–3):615–631MathSciNetzbMATHGoogle Scholar
  18. Suksompong W (2016) Assigning a small agreeable set of indivisible items to multiple players. In: Kambhampati S (ed) Proceedings of the twenty-fifth international joint conference on artificial intelligence, IJCAI 2016, New York, NY, USA, 9–15 July 2016, pp 489–495. IJCAI/AAAI PressGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS, PSL, LAMSADEUniversité Paris-DauphineParisFrance

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