Abstract
We study the computational complexity of envy minimization and maximizing the social welfare of graph-envy-free allocations in social networks. Besides the already known \(\mathrm {NP}\)-completeness of finding allocations with maximal utilitarian social welfare we prove that \(\mathrm {NP}\)-completeness is in general also given for the egalitarian social welfare and the Nash product. Moreover, we focus on an extended model, based on directed social relationship graphs and undirected social trading graphs, and analyze the computational complexity of reaching a graph-envy-free allocation by trades with so-called don’t care agents and without money.
This work was supported in part by DFG grant RO 1202/14-2.
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Notes
- 1.
Garey and Johnson [10] define a problem to be strongly \(\mathrm {NP}\) -complete if it is \(\mathrm {NP}\)-complete even when each of its numerical parameters is bounded by a polynomial in the length of the input. That implies that, unless \(\mathrm {P}= \mathrm {NP}\), strongly \(\mathrm {NP}\)-complete problems cannot have fully polynomial-time approximation schemes nor pseudo-polynomial-time algorithms.
- 2.
We need some notation to define this problem. Let \(v(\pi ) = (u_i(\pi _i))_{1 \le i \le n}\) be the utility vector induced by an allocation \(\pi \). Let \(v^*(\pi )\) be the vector that results from \(v(\pi )\) by sorting all entries nondecreasingly. In particular, \(v^*_1(\pi )\) is the utility of a worst-off agent and thus expresses egalitarian social welfare;
is the utility of a median-off agent; and \(v^*_n(\pi )\) is the utility of a best-off agent and thus expresses so-called elitist social welfare. - 3.
That the variant of this problem without don’t care agents is in \(\mathrm {NP}\) if the social trading network is a star can be seen as follows: Guess an allocation \(\pi \) from the given instance \(((A,R,\succ ),G_T, G_R, \pi _0, K)\), where \(G_T\) is a star, and check whether \(\pi \) is reachable from \(\pi _0\). Since \(G_T\) in particular is a tree, this can be done in polynomial time [11, Proposition 3]. Checking whether \(\pi \) is graph-envy-free can than also be done efficiently.
is the utility of a median-off agent; and References
Abdulkadiroǧlu, A., Sönmez, T.: Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica 66 (3), 689–701 (1998)
Aziz, H., Bouveret, S., Caragiannis, I., Giagkousi, I., Lang, J.: Knowledge, fairness, and social constraints. In: Proceedings of the 32nd AAAI Conference on Artificial Intelligence, pp. 4638–4645. AAAI Press (2018)
Bentert, M., Chen, J., Froese, V., Woeginger, G.J.: Good things come to those who swap objects on paths. arXiv:1905.04219 (2019)
Bergson, A.: A reformulation of certain aspects of welfare economics. Q. J. Econ. 52 (2), 310–334 (1938)
Bouveret, S., Cechlárová, K., Elkind, E., Igarashi, A., Peters, D.: Fair division of a graph. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence, pp. 135–141. AAAI Press/IJCAI (2017)
Bouveret, S., Chevaleyre, Y., Maudet, N.: Fair division of indivisible goods. In: Brandt, F., Conitzer, V., Endriss, U., Procaccia, A.D. (eds.) Handbook of Computational Social Choice, pp. 284–310. Cambridge University Press, Cambridge (2016)
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, pp. 283–291. IFAAMAS, July 2018
Chevaleyre, Y., et al.: Issues in multiagent resource allocation. Informatica 30 (1), 3–31 (2006)
Chevaleyre, Y., Endriss, U., Maudet, N.: Allocating goods on a graph to eliminate envy. In: Proceedings of the 22nd AAAI Conference on Artificial Intelligence, pp. 700–705. AAAI Press (2007)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)
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, pp. 213–219. AAAI Press/IJCAI (2017)
Heinen, T., Nguyen, N., Nguyen, T., Rothe, J.: Approximation and complexity of the optimization and existence problems for maximin share, proportional share, and minimax share allocation of indivisible goods. J. Auton. Agents Multi-Agent Syst. 32 (6), 741–778 (2018)
Heinen, T., Nguyen, N.-T., Rothe, J.: Fairness and rank-weighted utilitarianism in resource allocation. In: Walsh, T. (ed.) ADT 2015. LNCS (LNAI), vol. 9346, pp. 521–536. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-23114-3_31
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)
Lang, J., Rothe, J.: Fair division of indivisible goods. In: Rothe, J. (ed.) Economics and Computation. STBE, pp. 493–550. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-47904-9_8
Lorenz, M.O.: Methods of measuring the concentration of wealth. Publ. Am. Stat. Assoc. 9 (70), 209–219 (1905)
Papadimitriou, C.H.: Computational Complexity, 2nd edn. Addison-Wesley, Reading (1995)
Rothe, J.: Complexity Theory and Cryptology. An Introduction to Cryptocomplexity. EATCS Texts in Theoretical Computer Science. Springer, Heidelberg (2005). https://doi.org/10.1007/3-540-28520-2
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Lange, P., Rothe, J. (2019). Optimizing Social Welfare in Social Networks. In: Pekeč, S., Venable, K.B. (eds) Algorithmic Decision Theory. ADT 2019. Lecture Notes in Computer Science(), vol 11834. Springer, Cham. https://doi.org/10.1007/978-3-030-31489-7_6
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