Skip to main content

On Reachable Assignments in Cycles

  • 549 Accesses

Part of the Lecture Notes in Computer Science book series (LNAI,volume 13023)


The efficient and fair distribution of indivisible resources among agents is a common problem in the field of Multi-Agent-Systems. We consider a graph-based version of this problem called Reachable Assignment, introduced by Gourvès, Lesca, and Wilczynski [IJCAI, 2017]. The input for this problem consists of a set of agents, a set of objects, the agent’s preferences over the objects, a graph with the agents as vertices and edges encoding which agents can trade resources with each other, and an initial and a target distribution of the objects, where each agent owns exactly one object in each distribution. The question is then whether the target distribution is reachable via a sequence of rational trades. A trade is rational when the two participating agents are neighbors in the graph and both obtain an object they prefer over the object they previously held. We show that Reachable Assignment is solvable in \(\mathcal {O}(n^3)\) time when the input graph is a cycle with n vertices.


  • Multi-Agent Systems
  • Resource allocation
  • Polynomial-time algorithm
  • Reduction to 2-SAT

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD   69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   89.99
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


  1. 1.

    If \(\{\sigma _0^{-1}(p),\sigma ^{-1}(p)\} \in E \cap P_{\gamma }(q)\), then \(\xi _\gamma (p,q)\) contains two disjoint paths, one with \(\sigma _0^{-1}(p)\) as endpoint and one with \(\sigma ^{-1}(p)\).

  2. 2.

    Note that p is one of the m objects.

  3. 3.

    In this paper, we represent 2-SAT clauses by implications, equalities, and inequalities. Note that \(p \rightarrow q \equiv (\lnot p \vee q)\), \(p = q \equiv (p \vee q) \wedge (\lnot p \vee \lnot q)\), and \(p \ne q \equiv (p \vee \lnot q) \wedge (\lnot p \vee q)\).

  4. 4.

    We use the notation \(q \ne d(p,e)\) for some object q to avoid case distinctions. Since d(pe) is precomputed, this clause is equivalent to \(\lnot q\) if \(d(p,e)=1\) and q otherwise.


  1. Abraham, D.J., Blum, A., Sandholm, T.: Clearing algorithms for barter exchange markets: enabling nationwide kidney exchanges. In: Proceedings of the 8th ACM Conference on Electronic Commerce (EC ’07), pp. 295–304. ACM (2007)

    Google Scholar 

  2. Abraham, D.J., Cechlárová, K., Manlove, D.F., Mehlhorn, K.: Pareto optimality in house allocation problems. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 1163–1175. Springer, Heidelberg (2005).

    CrossRef  Google Scholar 

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

    Google Scholar 

  4. Bentert, M., Malík, J., Weller, M.: Tree containment with soft polytomies. In: Proceedings of the 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT ’18), LIPIcs, vol. 101, pp. 9:1–9:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

    Google Scholar 

  5. Beynier, A., et al.: Local envy-freeness in house allocation problems. Auton. Agents Multi-Agent Syst. 33(5), 591–627 (2019).

    CrossRef  Google Scholar 

  6. Brandt, F., Wilczynski, A.: On the convergence of swap dynamics to pareto-optimal matchings. In: Caragiannis, I., Mirrokni, V., Nikolova, E. (eds.) WINE 2019. LNCS, vol. 11920, pp. 100–113. Springer, Cham (2019).

    CrossRef  Google Scholar 

  7. Bredereck, R., Kaczmarczyk, A., Niedermeier, R.: Envy-free allocations respecting social networks. In: Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems (AAMAS ’18), pp. 283–291. International Foundation for Autonomous Agents and Multiagent Systems (ACM) (2018)

    Google Scholar 

  8. Cechlárová, K., Schlotter, I.: Computing the deficiency of housing markets with duplicate houses. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 72–83. Springer, Heidelberg (2010).

    CrossRef  Google Scholar 

  9. Chevaleyre, Y., Endriss, U., Maudet, N.: Allocating goods on a graph to eliminate envy. In: Proceedings of the 22nd Conference on Artificial Intelligence (AAAI ’07), pp. 700–705. AAAI Press (2007)

    Google Scholar 

  10. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  11. Gourvès, L., Lesca, J., Wilczynski, A.: Object allocation via swaps along a social network. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI ’17), pp. 213–219 (2017)

    Google Scholar 

  12. Gusfield, D., Wu, Y.: The three-state perfect phylogeny problem reduces to 2-SAT. Commun. Inf. Syst. 9(4), 295–302 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Huang, S., Xiao, M.: Object reachability via swaps under strict and weak preferences. Auton. Agents Multi-Agent Syst. 34(2), 1–33 (2020).

    CrossRef  MathSciNet  Google Scholar 

  14. Igarashi, A., Peters, D.: Pareto-optimal allocation of indivisible goods with connectivity constraints. In: Proceedings of the 33rd AAAI Conference on Artificial Intelligence (AAAI ’19), pp. 2045–2052. AAAI Press (2019)

    Google Scholar 

  15. Roth, A.E.: Incentive compatibility in a market with indivisible goods. Econ. Lett. 9(2), 127–132 (1982)

    CrossRef  MathSciNet  Google Scholar 

  16. Saffidine, A., Wilczynski, A.: Constrained swap dynamics over a social network in distributed resource reallocation. In: Deng, X. (ed.) SAGT 2018. LNCS, vol. 11059, pp. 213–225. Springer, Cham (2018).

    CrossRef  Google Scholar 

  17. Shapley, L., Scarf, H.: On cores and indivisibility. J. Math. Econ. 1(1), 23–37 (1974)

    CrossRef  MathSciNet  Google Scholar 

  18. Sönmez, T., Ünver, M.U.: House allocation with existing tenants: a characterization. Games Econ. Behav. 69(2), 425–445 (2010)

    CrossRef  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Matthias Bentert .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Müller, L., Bentert, M. (2021). On Reachable Assignments in Cycles. In: Fotakis, D., Ríos Insua, D. (eds) Algorithmic Decision Theory. ADT 2021. Lecture Notes in Computer Science(), vol 13023. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-87755-2

  • Online ISBN: 978-3-030-87756-9

  • eBook Packages: Computer ScienceComputer Science (R0)