Abstract
We consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preference over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small, since the problem turns out to be \(\mathsf {W}[3]\)-hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k.
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References
Aziz, H., Gaspers, S., Mackenzie, S., Walsh, T.: Fair assignment of indivisible objects under ordinal preferences. Artif. Intell. 227, 71–92 (2015)
Aziz, H., Schlotter, I., Walsh, T.: Control of fair division. In: IJCAI 2016, Proceedings of the 25th International Joint Conference on Artificial Intelligence, pp. 67–73 (2016)
Bartholdi, J.J., Tovey, C.A., Trick, M.A.: How hard is it to control an election? Math. Comput. Model. 16(8–9), 27–40 (1992)
Brams, S.J., Kilgour, D.M., Klamler, C.: Two-person fair division of indivisible items: An efficient, envy-free algorithm. Not. AMS 61(2), 130–141 (2014)
Chen, J., Huang, X., Kanj, I.A., Xia, G.: Strong computational lower bounds via parameterized complexity. J. Comput. Syst. Sci. 72(8), 1346–1367 (2006)
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London (2013)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series, vol. XIV. Springer, Berlin (2006)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)
Lipton, R.J., Markakis, E., Mossel, E., Saberi, A.: On approximately fair allocations of indivisible goods. In: EC 2004, Proceedings of the 5th ACM Conference on Electronic Commerce, pp. 125–131 (2004)
Mucha, M., Sankowski, P.: Maximum matchings via gaussian elimination. In: FOCS 2014, Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 248–255 (2004)
Nguyen, T., Vohra, R.: Near feasible stable matchings. In: EC 2015, Proceedings of the Sixteenth ACM Conference on Economics and Computation, pp. 41–42 (2015)
Schlotter, I., Dorn, B., de Haan, R.: Obtaining a proportional allocation by deleting items. CoRR, abs/1705.11060 (2017)
Segal-Halevi, E., Hassidim, A., Aumann, Y.: Waste makes haste: Bounded time protocols for envy-free cake cutting with free disposal. In: AAMAS 2014, Proceedings of the 14th International Conference on Autonomous Agents and Multi-Agent Systems, pp. 901–908 (2015)
Thulasiraman, K., Arumugam, S., Brandstädt, A., Nishizeki, T.: Handbook of Graph Theory, Combinatorial Optimization, and Algorithms. Chapman & Hall/CRC Computer and Information Science Series. CRC Press, Boca Raton (2015)
Acknowledgments
This work has been partly supported by COST Action IC1205 on Computational Social Choice, and has been supported by OTKA grants K108383 and K108947 and Austrian Science Fund grant J4047.
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Dorn, B., de Haan, R., Schlotter, I. (2017). Obtaining a Proportional Allocation by Deleting Items. In: Rothe, J. (eds) Algorithmic Decision Theory. ADT 2017. Lecture Notes in Computer Science(), vol 10576. Springer, Cham. https://doi.org/10.1007/978-3-319-67504-6_20
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DOI: https://doi.org/10.1007/978-3-319-67504-6_20
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