Autonomous Agents and Multi-Agent Systems

, Volume 33, Issue 5, pp 540–563 | Cite as

Chore division on a graph

  • Sylvain BouveretEmail author
  • Katarína Cechlárová
  • Julien Lesca


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.


Computational social choice Resource allocation Fair division Indivisible chores 



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.


  1. 1.
    Abebe, R., Kleinberg, J., & Parkes, D. C. (2017). Fair division via social comparison. In Proceedings of AAMAS’17 (pp. 281–289).Google Scholar
  2. 2.
    Aumann, Y., & Dombb, Y. (2010). The efficiency of fair division with connected pieces. In International workshop on internet and network economics (pp. 26–37). Springer.Google Scholar
  3. 3.
    Aziz, H., Rauchecker, G., Schryen, G., & Walsh, T. (2017). Algorithms for max-min share fair allocation of indivisible chores. In S. P. Singh & S. Markovitch (Eds.), Proceedings of the 31st AAAI conference on artificial intelligence (AAAI-17) (pp. 335–341). San Francisco, CA: AAAI Press.Google Scholar
  4. 4.
    Bei, X., Qiao, Y., & Zhang, S. (2017) Networked fairness in cake cutting. In Proceedings of the 26th international joint conference on artificial intelligence (IJCAI-17) (pp. 3632–3638).Google Scholar
  5. 5.
    Berman, P., Karpinski, M., & Scott, A. D. (2003). Approximation hardness of short symmetric instances of MAX-3SAT. Electronic Colloquium on Computational Complexity Report, number 49.Google Scholar
  6. 6.
    Bilò, V., Caragiannis, I., Flammini, M., Igarashi, A., Monaco, G., Peters, D., et al. (2018). Almost envy-free allocations with connected bundles. arXiv e-prints arXiv:1808.09406.
  7. 7.
    Bogomolnaia, A., Moulin, H., Sandomirskiy, F., & Yanovskaya, E. (2017). Competitive division of a mixed manna. Econometrica, 85(6), 1847–1871.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bouveret, S., Chevaleyre, Y., & Maudet, N. (2015). Chapter 12: Fair allocation of indivisible goods. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. D. Procaccia (Eds.), Handbook of computational social choice. Cambridge: Cambridge University Press.Google Scholar
  9. 9.
    Bouveret, S., Cechlárová, K., Elkind, E., Igarashi, A., & Peters, D. (2017). Fair division of a graph. In C. Sierra (Ed.), Proceedings of the 26th international joint conference on artificial intelligence (IJCAI-17), Melbourne, Australia (pp. 135–141), Scholar
  10. 10.
    Brams, S. J., Taylor, A., & Zwicker, W. S. (1997). A moving knife solution to the four-person envy-free cake division problem. Proceedings of the American Mathematical Society, 125, 547–554.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bredereck, R., Kaczmarczyk, A., & Niedermeier, R. (2018). Envy-free allocations respecting social networks. In Proceedings of the 17th international conference on autonomous agents and multiagent systems (AAMAS 2018) (pp. 1–9).Google Scholar
  12. 12.
    Budish, E. (2011). The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy, 119(6), 1061–1103.CrossRefGoogle Scholar
  13. 13.
    Caragiannis, I., Kaklamanis, C., Kanellopoulos, P., & Kyropoulou, M. (2012). The efficiency of fair division. Theory of Computing Systems, 50, 589–610.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cechlárová, K., & Pillárová, E. (2012). On the computability of equitable divisions. Discrete Optimization, 9, 249–257.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cechlárová, K., Doboš, J., & Pillárová, E. (2013). On the existence of equitable divisions. Information Sciences, 228, 239–245.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chevaleyre, Y., Endriss, U., & Maudet, N. (2007). Allocating goods on a graph to eliminate envy. In Proceedings of the 22nd AAAI conference on artificial intelligence (AAAI-07), Vancouver, British Columbia, Canada (pp. 700–705).Google Scholar
  17. 17.
    Dehghani, S., Farhadi, A., Hajiaghayi, M. T., & Yami, H. (2018). Envy-free chore division for an arbitrary number of agents. In Proceedings of the 29th annual ACM-SIAM symposium on discrete algorithms, SODA 2018, New Orleans, LA, USA, January 7–10, 2018 (pp. 2564–2583).Google Scholar
  18. 18.
    Demko, S., & Hill, T. P. (1998). Equitable distribution of indivisible items. Mathematical Social Sciences, 16, 145–158.CrossRefzbMATHGoogle Scholar
  19. 19.
    Dinits, E. A. (1970). Algoritm resheniya zadachi o maximaĺnom potoke v seti so stepennoǐ otsenkoǐ (Russian). Doklady Akademii Nauk SSSR, 194, 754–757.MathSciNetGoogle Scholar
  20. 20.
    Farhadi, A., & Hajiaghayi, M. (2018). On the complexity of chore division. In J. Lang (Ed.), Proceedings of the 27th international joint conference on artificial intelligence (IJCAI-18), Stockholm, Sweden (pp. 226–232). Scholar
  21. 21.
    Gardner, M. (1978). Aha! Insight. New York: Freeman.Google Scholar
  22. 22.
    Garey, M. R., & Johnson, D. S. (1979). Computers and intractability, a guide to the theory of NP-completeness. New York: Freeman.zbMATHGoogle Scholar
  23. 23.
    Gourvès, L., Lesca, J., & Wilczynski, A. (2017). Object allocation via swaps along a social network. In C. Sierra (Ed.), Proceedings of the 26th international joint conference on artificial intelligence (IJCAI-17), Melbourne, Australia (pp. 213–219), Scholar
  24. 24.
    Heydrich, S., & van Stee, R. (2015). Dividing connected chores fairly. Theoretical Computer Science, 593, 51–61.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Igarashi, A., & Peters, D. (2018). Pareto-optimal allocation of indivisible goods with connectivity constraints. Accepted to AAAI-19.Google Scholar
  26. 26.
    Lipton, R., Markakis, E., Mossel, E., & Saberi, A. (2004). On approximately fair allocations of indivisible goods. In Proceedings of EC’04.Google Scholar
  27. 27.
    Lonc, Z., & Truszczynski, M. (2018). Maximin share allocations on cycles. In J. Lang (Ed.), Proceedings of the 27th international joint conference on artificial intelligence (IJCAI-18), Stockholm, Sweden (pp. 410–416), Scholar
  28. 28.
    Markakis, E., & Psomas, C.-A. (2011). On worst-case allocations in the presence of indivisible goods. In N. Chen, E. Elkind, & E. Koutsoupias (Eds.), Internet and network economics—7th international workshop, WINE 2011, Singapore, December 11–14, 2011. Proceedings, volume 7090 of Lecture Notes in Computer Science. Springer.Google Scholar
  29. 29.
    Oh, H., Procaccia, A., & Suksompong, W. (2018). Fairly allocating many goods with few queries. arXiv:1807.11367.
  30. 30.
    Peterson, E., & Su, F. E. (2002). Four-person envy-free chore division. Mathematics Magazine, 75(2), 117–122.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Robertson, J., & Webb, W. (1998). Cake-cutting algorithms: Be fair if you can. Natick: AK Peters/CRC Press.CrossRefzbMATHGoogle Scholar
  32. 32.
    Rothe, J. (Ed.). (2015). Economics and computation: An introduction to algorithmic game theory, computational social choice, and fair division. Berlin: Springer.zbMATHGoogle Scholar
  33. 33.
    Schrijver, A. (2003). Combinatorial optimization: Polyhedra and efficiency. Berlin: Springer.zbMATHGoogle Scholar
  34. 34.
    Steinhaus, H. (1948). The problem of fair division. Econometrica, 16, 101–104.Google Scholar
  35. 35.
    Stromquist, W. (1980). How to cut a cake fairly. American Mathematical Monthly, 87, 640–644.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Stromquist, W. (2008). Envy-free divisions cannot be found by finite protocols. The Electronic Journal of Combinatorics, 15, 145–158.MathSciNetzbMATHGoogle Scholar
  37. 37.
    Suksompong, W. (2017). Fairly allocating contiguous blocks of indivisible items. In Proceedings of the 10th international symposium on algorithmic game theory (SAGT’17) (pp. 333–344).Google Scholar
  38. 38.
    Woodall, D. R. (1980). Dividing a cake fairly. Journal of Mathematical Analysis and Applications, 78, 233–247.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CNRS, Grenoble INP, LIGUniv. Grenoble AlpesGrenobleFrance
  2. 2.P.J. Šafárik UniversityKosiceSlovakia
  3. 3.PSL, CNRS, LAMSADEUniversité Paris-DauphineParisFrance

Personalised recommendations