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Approximation and complexity of the optimization and existence problems for maximin share, proportional share, and minimax share allocation of indivisible goods

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Abstract

This paper is concerned with various types of allocation problems in fair division of indivisible goods, aiming at maximin share, proportional share, and minimax share allocations. However, such allocations do not always exist, not even in very simple settings with two or three agents. A natural question is to ask, given a problem instance, what is the largest value c for which there is an allocation such that every agent has utility of at least c times her fair share. We first prove that the decision problem of checking if there exists a minimax share allocation for a given problem instance is \(\mathrm {NP}\)-hard when the agents’ utility functions are additive. We then show that, for each of the three fairness notions, one can approximate c by a polynomial-time approximation scheme, assuming that the number of agents is fixed. Next, we investigate the restricted cases when utility functions have values in \(\{0,1\}\) only or are defined based on scoring vectors (Borda and lexicographic vectors), and we obtain several tractability results for these cases. Interestingly, we show that maximin share allocations can always be found efficiently with Borda utilities, which cannot be guaranteed for general additive utilities. In the nonadditive setting, we show that there exists a problem instance for which there is no c-maximin share allocation, for any constant c. We explore a class of symmetric submodular utilities for which there exists a tight \(\frac{1}{2}\)-maximin share allocation, and show how it can be approximated to within a factor of \(\nicefrac {1}{4}\).

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Notes

  1. A PTAS for Maximin-Opt has already been given by Aziz et al. [5].

  2. Lorenz dominance (see, e.g, Golden and Perny [34], Endriss [28]) is related to majorization and it can be used when utility functions are not identical.

  3. Alternatively, one can model the problem as a max-flow problem and use a known polynomial-time algorithm for solving it.

  4. Note that Darmann and Klamler [25] consider Borda vectors of the form \((m-1,\ldots ,1,0)\), which slightly differs from our Borda vectors, \((m,\ldots ,2,1)\), which can be seen as an additive translation of the former one. However, as shown by Darmann and Klamler [25], proportionality is unchanged under such a translation.

  5. Note that in our original definition of lexicographic utilities in Sect. 6.2, it is required that k must be equal to j.

  6. We thank an anonymous reviewer for pointing us to this example, which is simpler than the one we used in the conference version [54].

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Acknowledgements

We thank the anonymous JAAMAS, ADT 2015, and AAMAS 2017 reviewers for their helpful comments. We also thank Khaled Elbassioni for very helpful discussions. This work was supported in part by DFG Grant RO 1202/14-2, and by Vietnam National Foundation for Science and Technology Development (NAFOSTED Project No. 102.01-2015.33).

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Authors and Affiliations

  1. Saarland University and IMPRS-CS, Saarland Informatics Campus, 66123, Saarbrücken, Germany

    Tobias Heinen

  2. Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225, Düsseldorf, Germany

    Nhan-Tam Nguyen & Jörg Rothe

  3. Hai Phong University, Hai Phong, Vietnam

    Trung Thanh Nguyen

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  1. Tobias Heinen
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Preliminary versions of parts of this paper appear in the proceedings of the 4th International Conference on Algorithmic Decision Theory [40] and of the 16th International Conference on Autonomous Agents and Multiagent Systems [54]. This paper presents novel results that are not contained in [40, 54], including Theorems 2, 5, and 9 and its corollaries. Also, it contains the missing proof of Theorem 10 and of Proposition 7, which were not given in [54] due to space constraints.

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Heinen, T., Nguyen, NT., Nguyen, T.T. et al. Approximation and complexity of the optimization and existence problems for maximin share, proportional share, and minimax share allocation of indivisible goods. Auton Agent Multi-Agent Syst 32, 741–778 (2018). https://doi.org/10.1007/s10458-018-9393-0

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