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Maximin Shares Under Cardinality Constraints

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Multi-Agent Systems (EUMAS 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13442))

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Abstract

We study the problem of fair allocation of a set of indivisible items among agents with additive valuations, under cardinality constraints. In this setting, the items are partitioned into categories, each with its own limit on the number of items it may contribute to any bundle. We consider the fairness measure known as the maximin share (MMS) guarantee, and propose a novel polynomial-time algorithm for finding 1/2-approximate MMS allocations for goods—an improvement from the previously best available guarantee of 11/30. For single-category instances, we show that a modified variant of our algorithm is guaranteed to produce 2/3-approximate MMS allocations. Among various other existence and non-existence results, we show that a \((\sqrt{n}/(2\sqrt{n} - 1))\)-approximate MMS allocation always exists for goods. For chores, we show similar results as for goods, with a 2-approximate algorithm in the general case and a 3/2-approximate algorithm for single-category instances. We extend the notions and algorithms related to ordered and reduced instances to work with cardinality constraints, and combine these with bag filling style procedures to construct our algorithms.

A preliminary version of this paper appeared at AAMAS 2022 as an extended abstract [21].

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Notes

  1. 1.

    Instances with more than \(nk_h\) items in category \(C_h\) can be handled by ordering the instance (see Sect. 3) and ignoring the worst items in the category.

  2. 2.

    The definition is equivalent for chores.

  3. 3.

    We use l instead of the usual \(\ell \) to avoid conflicting use of symbols.

  4. 4.

    In the unconstrained setting, a PTAS exists for finding the MMS of each individual agent [29], but this PTAS does not extend to fair allocation under cardinality constraints and there does not, to our knowledge, exist a PTAS for this problem.

  5. 5.

    Available at https://arxiv.org/abs/2106.07300, together with an earlier preprint version (v1) containing some preliminary experiments and corresponding source code.

  6. 6.

    If \(v_i(M) = 0\), normalization does not work. However, since this implies \(\mu _i = 0\), Corollary 1 can be used to eliminate agent i from the instance.

  7. 7.

    The term reduction here refers to data reduction, as the term is used in parameterized algorithm design, rather than to the problem transformations of complexity theory.

  8. 8.

    See Example 1 in the extended version for a simple instance where this fails.

  9. 9.

    As with goods, normalization does not work if \(v_i(M) = 0\). In this case, i can be removed from the (ordered) instance by allocating i the \(k_h\) worst chores in each \(C_h\). This would constitute a valid reduction.

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Acknowledgment

The authors wish to acknowledge valuable input from anonymous reviewers.

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  1. Norwegian University of Science and Technology, Trondheim, Norway

    Halvard Hummel & Magnus Lie Hetland

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  1. University of DĂĽsseldorf, DĂĽsseldorf, Germany

    Dorothea Baumeister

  2. University of DĂĽsseldorf, DĂĽsseldorf, Germany

    Jörg Rothe

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Hummel, H., Hetland, M.L. (2022). Maximin Shares Under Cardinality Constraints. In: Baumeister, D., Rothe, J. (eds) Multi-Agent Systems. EUMAS 2022. Lecture Notes in Computer Science(), vol 13442. Springer, Cham. https://doi.org/10.1007/978-3-031-20614-6_11

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