Improving Approximation Guarantees for Maximin Share

Hannaneh Akrami, Jugal Garg, Eklavya Sharma, Setareh Taki

Abstract: We consider fair division of a set of indivisible goods among n agents with additive valuations using the 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 (1-out-of-n) MMS value. An allocation is called MMS if all agents receive their MMS values. However, since MMS allocations do not always exist [Kurokawa et al., JACM'18], the focus shifted to investigating its ordinal and multiplicative approximations.In the ordinal approximation, the goal is to show the existence of 1-out-of-d MMS allocations (for the smallest possible d > n). A series of works led to the state-of-the-art factor of d = ⌊3n/2⌋ [Hosseini et al., JAIR'21]. We show that 1--out-of-4⌈n/3⌉ MMS allocations always exist, thereby improving the state-of-the-art of ordinal approximation. In the multiplicative approximation, the goal is to show the existence of α-MMS allocations (for the largest possible α < 1), which guarantees each agent at least α times her MMS value. A series of works in the last decade led to the state-of-the-art factor of [EQUATION] [Akrami and Garg, SODA'24].We introduce a general framework of approximate MMS with agent priority ranking. We order the agents, and agents earlier in the order are considered more important. An allocation is said to be T-MMS, for a non-increasing sequence T := (τ1,...,τn) of numbers, if the agent at rank i in the order gets a bundle of value at least τi times her MMS value. This framework captures both ordinal approximation and multiplicative approximation as special cases. We show the existence of T-MMS allocations where [EQUATION] for all i. Furthermore, by ordering the agents randomly, we can get allocations that are [EQUATION](3/4 + 1/12n)-MMS ex-post and [EQUATION]-MMS ex-ante. We also investigate the limitations of our algorithm and show that it does not give better than (0.8631 + 1/2n)-MMS ex-ante.The full version of this paper is available at https://arxiv.org/abs/2307.12916v2.