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Obtaining a Proportional Allocation by Deleting Items

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Algorithmic Decision Theory (ADT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10576))

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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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Notes

  1. 1.

    For a complete proof of the correctness of their algorithm, see also [2].

  2. 2.

    We present Theorem 1 so that we can re-use its proof for Theorems 3 and 4.

  3. 3.

    Here, we use an effective variant of “little o” (see, e.g. [7, Definition 3.22]).

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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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Authors and Affiliations

  1. University of Tübingen, Tübingen, Germany

    Britta Dorn

  2. University of Amsterdam, Amsterdam, The Netherlands

    Ronald de Haan

  3. Budapest University of Technology and Economics, Budapest, Hungary

    Ildikó Schlotter

Authors
  1. Britta Dorn
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  2. Ronald de Haan
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  3. Ildikó Schlotter
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Corresponding author

Correspondence to Ildikó Schlotter .

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Editors and Affiliations

  1. Heinrich Heine University, Düsseldorf, Germany

    Jörg Rothe

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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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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67503-9

  • Online ISBN: 978-3-319-67504-6

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