Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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.
Note that p is one of the m objects.
- 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.
We use the notation \(q \ne d(p,e)\) for some object q to avoid case distinctions. Since d(p, e) is precomputed, this clause is equivalent to \(\lnot q\) if \(d(p,e)=1\) and q otherwise.
References
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)
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). https://doi.org/10.1007/11602613_115
Bentert, M., Chen, J., Froese, V., Woeginger, G.J.: Good things come to those who swap objects on paths. CoRR abs/1905.04219 (2019)
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)
Beynier, A., et al.: Local envy-freeness in house allocation problems. Auton. Agents Multi-Agent Syst. 33(5), 591–627 (2019). https://doi.org/10.1007/s10458-019-09417-x
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). https://doi.org/10.1007/978-3-030-35389-6_8
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)
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). https://doi.org/10.1007/978-3-642-17493-3_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)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012)
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)
Gusfield, D., Wu, Y.: The three-state perfect phylogeny problem reduces to 2-SAT. Commun. Inf. Syst. 9(4), 295–302 (2009)
Huang, S., Xiao, M.: Object reachability via swaps under strict and weak preferences. Auton. Agents Multi-Agent Syst. 34(2), 1–33 (2020). https://doi.org/10.1007/s10458-020-09477-4
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)
Roth, A.E.: Incentive compatibility in a market with indivisible goods. Econ. Lett. 9(2), 127–132 (1982)
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). https://doi.org/10.1007/978-3-319-99660-8_19
Shapley, L., Scarf, H.: On cores and indivisibility. J. Math. Econ. 1(1), 23–37 (1974)
Sönmez, T., Ünver, M.U.: House allocation with existing tenants: a characterization. Games Econ. Behav. 69(2), 425–445 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
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. https://doi.org/10.1007/978-3-030-87756-9_18
Download citation
DOI: https://doi.org/10.1007/978-3-030-87756-9_18
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)