Maximin Shares Under Cardinality Constraints

Hummel, Halvard; Hetland, Magnus Lie

doi:10.1007/978-3-031-20614-6_11

Cited by 4 publications

(2 citation statements)

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“…In addition, a polynomial time O(mlogm + n) algorithm was designed for the job scheduling problem, and the optimal scheduling scheme of 11 9 -approximate was obtained. Under cardinality constraints, Hummel and Hetland (2022) [93] proved that there is a 2-approximate MMS allocation in general and that there is a 3 2 -approximate MMS allocation in single-category instances. Feige and Norkin (2022) [94] considered the problem of approximate MMS allocation between three agents with additive disutility and proved that there is always a 19 18 -approximate MMS allocation.…”

Section: Maximin Share Fairnessmentioning

confidence: 99%

A Survey on Fair Allocation of Chores

Guo,

Li,

Deng

2023

Mathematics

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Wherever there is group life, there has been a social division of labor and resource allocation, since ancient times. Examples include ant colonies, bee colonies, and wolf colonies. Different roles are responsible for different tasks. The same is true of human beings. Human beings are the largest social group in nature, among whom there are intricate social networks and interest networks between individuals. In such a complex relationship, how do decision makers allocate resources or tasks to individuals in a fair way? This is a topic worthy of further study. In recent decades, fair allocation has been at the core of research in economics, mathematics and other fields. The fair allocation problem is to assign a set of items to a set of agents so that each agent’s allocation is as fair as possible to satisfy each agent. The fairness measurements followed in current research include envy-freeness, proportionality, equitability, maximin share fairness, competitive equilibrium, maximum Nash social diswelfare, and so on. In this paper, the main concern is the allocation of chores. We discuss this problem in two parts: divisible and indivisible. We comprehensively review the existing results, algorithms, and approximations that meet various fairness criteria in chronological order. The relevant results of achieving fairness and efficiency are also discussed. In addition, we propose some open questions and future research directions for this problem based on existing research.

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Section: Maximin Share Fairnessmentioning

confidence: 99%

A Survey on Fair Allocation of Chores

Guo,

Li,

Deng

2023

Mathematics

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“…Hummel et. al [24] improved the existence guarantee to 2 3 and proposed a 2 3 -approximation guaranteed polynomial time algorithm.…”

Section: Related Workmentioning

confidence: 99%

The budgeted maximin share allocation problem

Li,

Deng

2023

Preprint

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We are given a set of indivisible goods and a set of n agents where each good has a size and each agent has an additive valuation function and a budget. The budgeted maximin share allocation problem is to find a feasible allocation such that the size of the bundle allocated to each agent does not exceed its budget, and the minimum ratio of the valuation and the maximin share (MMS) value of any agent is as large as possible, where the MMS value of each agent is that he can achieve by dividing the goods into n bundles, and receiving his least desirable bundle. In this paper, we prove that the existence of 1/3 -approximate MMS allocation and give an instance which does have a (3 /4 + ε)-approximate MMS allocation, for any ε ∈ (0, 1). Moreover, we provide a polynomial time algorithm to find an ( 1/3 − ε)-approximate MMS allocation, and prove that there is no polynomial time algorithm which can find a ( 2/3 + ε)-approximate MMS allocation for any instance unless P = N P.

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