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.
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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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