9 research outputs found

    The Fair Division of Hereditary Set Systems

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    We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form an hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least 0.36670.3667 times the maximin share of the agent. This improves upon the current best known guarantee of 0.20.2 due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most 0.37380.3738. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.Comment: 22 pages, 1 figure, full version of WINE 2018 submissio

    Simplification and Improvement of MMS Approximation

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    We consider the problem of fairly allocating a set of indivisible goods among nn agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The current best approximation factor, for which the existence is known, is (34+112n)(\frac{3}{4} + \frac{1}{12n}) [Garg and Taki, 2021]. Most of these results are based on complicated analyses, especially those providing better than 2/32/3 factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of (34+min(136,316n4))(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4})). For small nn, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques

    Randomized and Deterministic Maximin-share Approximations for Fractionally Subadditive Valuations

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    We consider the problem of guaranteeing maximin-share (MMS) when allocating a set of indivisible items to a set of agents with fractionally subadditive (XOS) valuations. For XOS valuations, it has been previously shown that for some instances no allocation can guarantee a fraction better than 1/21/2 of maximin-share to all the agents. Also, a deterministic allocation exists that guarantees 0.2192250.219225 of the maximin-share of each agent. Our results involve both deterministic and randomized allocations. On the deterministic side, we improve the best approximation guarantee for fractionally subadditive valuations to 3/13=0.2307693/13 = 0.230769. We develop new ideas on allocating large items in our allocation algorithm which might be of independent interest. Furthermore, we investigate randomized algorithms and the Best-of-both-worlds fairness guarantees. We propose a randomized allocation that is 1/41/4-MMS ex-ante and 1/81/8-MMS ex-post for XOS valuations. Moreover, we prove an upper bound of 3/43/4 on the ex-ante guarantee for this class of valuations

    Improving Approximation Guarantees for Maximin Share

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    We consider fair division of a set of indivisible goods among nn agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her MMS value. An allocation is called MMS if all agents receive their MMS values. However, since MMS allocations do not always exist, the focus shifted to investigating its ordinal and multiplicative approximations. In the ordinal approximation, the goal is to show the existence of 11-out-of-dd MMS allocations (for the smallest possible d>nd>n). A series of works led to the state-of-the-art factor of d=3n/2d=\lfloor 3n/2 \rfloor [HSSH21]. We show that 11-out-of-4n/3\lceil 4n/3\rceil MMS allocations always exist. In the multiplicative approximation, the goal is to show the existence of α\alpha-MMS allocations (for the largest possible α<1\alpha < 1) which guarantees each agent at least α\alpha times her MMS value. A series of works in the last decade led to the state-of-the-art factor of α=34+33836\alpha = \frac{3}{4} + \frac{3}{3836} [AG23]. We introduce a general framework of (α,β,γ)(\alpha, \beta, \gamma)-MMS that guarantees α\alpha fraction of agents β\beta times their MMS values and the remaining (1α)(1-\alpha) fraction of agents γ\gamma times their MMS values. The (α,β,γ)(\alpha, \beta, \gamma)-MMS captures both ordinal and multiplicative approximations as its special cases. We show that (2(1β)/β,β,3/4)(2(1 -\beta)/\beta, \beta, 3/4)-MMS allocations always exist. Furthermore, since we can choose the 2(1β)/β2(1-\beta)/\beta fraction of agents arbitrarily in our algorithm, this implies (using β=3/2\beta=\sqrt{3}/2) the existence of a randomized allocation that gives each agent at least 3/4 times her MMS value (ex-post) and at least (17324)/43>0.785(17\sqrt{3} - 24)/4\sqrt{3} > 0.785 times her MMS value in expectation (ex-ante)

    Breaking the 3/43/4 Barrier for Approximate Maximin Share

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    We study the fundamental problem of fairly allocating a set of indivisible goods among nn agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her MMS value. An allocation is called MMS if all agents receive at least their MMS value. However, since MMS allocations need not exist when n>2n>2, a series of works showed the existence of approximate MMS allocations with the current best factor of 34+O(1n)\frac{3}{4} + O(\frac{1}{n}). The recent work by Akrami et al. showed the limitations of existing approaches and proved that they cannot improve this factor to 3/4+Ω(1)3/4 + \Omega(1). In this paper, we bypass these barriers to show the existence of (34+33836)(\frac{3}{4} + \frac{3}{3836})-MMS allocations by developing new reduction rules and analysis techniques

    Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results

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    Fair resource allocation is an important problem in many real-world scenarios, where resources such as goods and chores must be allocated among agents. In this survey, we delve into the intricacies of fair allocation, focusing specifically on the challenges associated with indivisible resources. We define fairness and efficiency within this context and thoroughly survey existential results, algorithms, and approximations that satisfy various fairness criteria, including envyfreeness, proportionality, MMS, and their relaxations. Additionally, we discuss algorithms that achieve fairness and efficiency, such as Pareto Optimality and Utilitarian Welfare. We also study the computational complexity of these algorithms, the likelihood of finding fair allocations, and the price of fairness for each fairness notion. We also cover mixed instances of indivisible and divisible items and investigate different valuation and allocation settings. By summarizing the state-of-the-art research, this survey provides valuable insights into fair resource allocation of indivisible goods and chores, highlighting computational complexities, fairness guarantees, and trade-offs between fairness and efficiency. It serves as a foundation for future advancements in this vital field

    The Fair Division of Hereditary Set Systems

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