Skip to main content
SpringerLink
Log in

Object reachability via swaps under strict and weak preferences

  • Published:
Autonomous Agents and Multi-Agent Systems Aims and scope Submit manuscript

Abstract

The Housing Market problem is a widely studied resource allocation problem. In this problem, each agent can only receive a single object and has preferences over all objects. Starting from an initial endowment, we want to reach a certain assignment via a sequence of rational trades. We first consider whether an object is reachable for a given agent under a social network, where a trade between two agents is allowed if they are neighbors in the network and no participant has a deficit from the trade. Assume that the preferences of the agents are strict (no tie among objects is allowed). This problem is polynomial-time solvable in a star-network and NP-complete in a tree-network. It is left as a challenging open problem whether the problem is polynomial-time solvable when the network is a path. We answer this open problem positively by giving a polynomial-time algorithm. Then we show that when the preferences of the agents are weak (ties among objects are allowed), the problem becomes NP-hard when the network is a path and can be solved in polynomial time when the network is a star. Besides, we consider the computational complexity of finding different optimal assignments for the problem in the special case where the network is a path or a star.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Gärdenfors, P. (1973). Assignment problem based on ordinal preferences. Management Science, 20(3), 331–340.

    Article  Google Scholar 

  2. Wilson, L. B. (1977). Assignment using choice lists. Journal of the Operational Research Society, 28(3), 569–578.

    Article  MathSciNet  Google Scholar 

  3. Abdulkadiroğlu, A., & Sönmez, T. (1998). Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica, 66(3), 689–701.

    Article  MathSciNet  Google Scholar 

  4. Manlove, D. F. (2013). Algorithmics of matching under preferences (Vol. 2). Singapore: World Scientific.

    Book  Google Scholar 

  5. Shapley, L., & Scarf, H. (1974). On cores and indivisibility. Journal of Mathematical Economics, 1(1), 23–37.

    Article  MathSciNet  Google Scholar 

  6. Abdulkadiroğlu, A., & Sönmez, T. (1999). House allocation with existing tenants. Journal of Economic Theory, 88(2), 233–260.

    Article  Google Scholar 

  7. Roth, A. E., Sönmez, T., & Ünver, M. U. (2004). Kidney exchange. Quarterly Journal of Economics, 119, 457–488.

    Article  Google Scholar 

  8. Alcalde-Unzu, J., & Molis, E. (2011). Exchange of indivisible goods and indifferences: The top trading absorbing sets mechanisms. Games and Economic Behavior, 73(1), 1–16.

    Article  MathSciNet  Google Scholar 

  9. Jaramillo, P., & Manjunath, V. (2012). The difference indifference makes in strategy-proof allocation of objects. Journal of Economic Theory, 147(5), 1913–1946.

    Article  MathSciNet  Google Scholar 

  10. Aziz, H., & De Keijzer, B. (2012). Housing markets with indifferences: A tale of two mechanisms. In Proceedings of the twenty-sixth AAAI conference on artificial intelligence (pp. 1249–1255).

  11. Sabán, D., & Sethuraman, J. (2013). House allocation with indifferences: A generalization and a unified view. In Proceedings of the fourteenth ACM conference on electronic commerce (pp. 803–820).

  12. Ehlers, L. (2014). Top trading with fixed tie-breaking in markets with indivisible goods. Journal of Economic Theory, 151, 64–87.

    Article  MathSciNet  Google Scholar 

  13. Sonoda, A., Fujita, E., Todo, T., & Yokoo, M. (2014).Two case studies for trading multiple indivisible goods with indifferences. In Proceedings of the twenty-eighth AAAI conference on artificial intelligence (pp. 791–797).

  14. Ahmad, G. (2017). Essays on housing market problem. Ph.D. Thesis, Texas A & M University.

  15. Gourvès, L., Lesca, J., & Wilczynski, A. (2017). Object allocation via swaps along a social network. In Proceedings of the twenty-sixth international joint conference on artificial intelligence (pp. 213–219).

  16. Abebe, R., Kleinberg, J. M., & Parkes, D. C. (2017). Fair division via social comparison. In Proceedings of the sixteenth conference on autonomous agents and multiagent systems (pp. 281–289).

  17. Bei, X., Qiao, Y., & Zhang, S. (2017). Networked fairness in cake cutting. In Proceedings of the twenty-sixth international joint conference on artificial intelligence (pp. 3632–3638).

  18. Bredereck, R., Kaczmarczyk, A., & Niedermeier, R. (2018). Envy-free allocations respecting social networks. In Proceedings of the seventeenth international conference on autonomous agents and multiagent systems (pp. 283–291).

  19. Beynier, A., Chevaleyre, Y., Gourvès, L., Lesca, J., Maudet, N., & Wilczynski, A. (2018). Local envy-freeness in house allocation problems. In Proceedings of the seventeenth international conference on autonomous agents and multiagent systems (pp. 292–300).

  20. Müller, L., & Bentert, M. (2020). On reachable assignments in cycles and cliques. CoRR, arXiv:abs/2005.02218.

  21. Damamme, A., Beynier, A., Chevaleyre, Y., & Maudet, N. (2015). The power of swap deals in distributed resource allocation. In Proceedings of the 2015 international conference on autonomous agents and multiagent systems (pp. 625–633).

  22. Bentert, M., Chen, J., Froese, V., & Woeginger, G. J. (2019). Good things come to those who swap objects on paths. CoRR, arXiv:abs/1905.04219.

  23. Huang, S., & Xiao, M. (2019). Object reachability via swaps along a line. In Proceedings of the thirty-third AAAI conference on artificial intelligence (pp. 2037–2044).

  24. Aspvall, B., Plass, M. F., & Tarjan, R. E. (1979). A linear-time algorithm for testing the truth of certain quantified boolean formulas. Information Processing Letters, 8(3), 121–123.

    Article  MathSciNet  Google Scholar 

  25. Yoshinaka, R. (2005). Higher-order matching in the linear lambda calculus in the absence of constants is NP-complete. In Proceedings of the sixteenth international conference on rewriting techniques and applications (pp. 235–249).

  26. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: W. H. Freeman.

    MATH  Google Scholar 

  27. Mak-Hau, V. H. (2017). On the kidney exchange problem: Cardinality constrained cycle and chain problems on directed graphs: a survey of integer programming approaches. Journal of Combinatorial Optimization, 33(1), 35–59.

    Article  MathSciNet  Google Scholar 

  28. Saffidine, A., & Wilczynski, A. (2018). Constrained swap dynamics over a social network in distributed resource reallocation. In Algorithmic game theory—11th international symposium, SAGT 2018, Beijing, China, September 11–14, 2018, Proceedings (pp. 213–225).

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China, under Grants 61972070 and 61772115.

Author information

Authors and Affiliations

  1. University of Electronic Science and Technology of China, Chengdu, China

    Sen Huang & Mingyu Xiao

Authors
  1. Sen Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingyu Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingyu Xiao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, S., Xiao, M. Object reachability via swaps under strict and weak preferences. Auton Agent Multi-Agent Syst 34, 51 (2020). https://doi.org/10.1007/s10458-020-09477-4

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s10458-020-09477-4

Keywords

Search

Navigation