Approximation Algorithms for Computing Maximin Share Allocations

We study the problem of computing maximin share allocations, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of an agent is the best she can guarantee to herself, if she is allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then is to find a partition, where each agent is guaranteed her maximin share. Such allocations do not always exist, hence we resort to approximation algorithms. Our main result is a 2/3-approximation that runs in polynomial time for any number of agents and goods. This improves upon the algorithm of Procaccia and Wang (2014), which is also a 2/3-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of the algorithm in Procaccia and Wang (2014), exploiting the construction of carefully selected matchings in a bipartite graph representation of the problem. Furthermore, motivated by the apparent difficulty in establishing lower bounds, we undertake a probabilistic analysis. We prove that in randomly generated instances, maximin share allocations exist with high probability. This can be seen as a justification of previously reported experimental evidence. Finally, we provide further positive results for two special cases arising from previous works. The first is the intriguing case of three agents, where we provide an improved 7/8-approximation. The second case is when all item values belong to {0, 1, 2}, where we obtain an exact algorith

Asymptotic Existence of Proportionally Fair Allocations

Fair division has long been an important problem in the economics literature. In this note, we consider the existence of proportionally fair allocations of indivisible goods, i.e., allocations of indivisible goods in which every agent gets at least her proportionally fair share according to her own utility function. We show that when utilities are additive and utilities for individual goods are drawn independently at random from a distribution, proportionally fair allocations exist with high probability if the number of goods is a multiple of the number of agents or if the number of goods grows asymptotically faster than the number of agents

Computing an Approximately Optimal Agreeable Set of Items

We study the problem of finding a small subset of items that is \emph{agreeable} to all agents, meaning that all agents value the subset at least as much as its complement. Previous work has shown worst-case bounds, over all instances with a given number of agents and items, on the number of items that may need to be included in such a subset. Our goal in this paper is to efficiently compute an agreeable subset whose size approximates the size of the smallest agreeable subset for a given instance. We consider three well-known models for representing the preferences of the agents: ordinal preferences on single items, the value oracle model, and additive utilities. In each of these models, we establish virtually tight bounds on the approximation ratio that can be obtained by algorithms running in polynomial time.Comment: A preliminary version appeared in Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI), 201

Maximin Fairness with Mixed Divisible and Indivisible Goods

We study fair resource allocation when the resources contain a mixture of divisible and indivisible goods, focusing on the well-studied fairness notion of maximin share fairness (MMS). With only indivisible goods, a full MMS allocation may not exist, but a constant multiplicative approximate allocation always does. We analyze how the MMS approximation guarantee would be affected when the resources to be allocated also contain divisible goods. In particular, we show that the worst-case MMS approximation guarantee with mixed goods is no worse than that with only indivisible goods. However, there exist problem instances to which adding some divisible resources would strictly decrease the MMS approximation ratio of the instance. On the algorithmic front, we propose a constructive algorithm that will always produce an $\alpha$-MMS allocation for any number of agents, where $\alpha$ takes values between $1/2$ and $1$ and is a monotone increasing function determined by how agents value the divisible goods relative to their MMS values.Comment: Appears in the 35th AAAI Conference on Artificial Intelligence (AAAI), 202

Multiple Birds with One Stone: Beating 1/21/2 for EFX and GMMS via Envy Cycle Elimination

Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a $(\phi-1)$-approximation of envy-freeness up to any good (EFX) and a $\frac{2}{\phi +2}$-approximation of groupwise maximin share fairness (GMMS), where $\phi$ is the golden ratio ($\phi \approx 1.618$). The best known approximation factor for either one of these fairness notions prior to this work was $1/2$. Moreover, the returned allocation achieves envy-freeness up to one good (EF1) and a $2/3$-approximation of pairwise maximin share fairness (PMMS). While EFX is our primary focus, we also exhibit how to fine-tune our algorithm and improve the guarantees for GMMS or PMMS. Finally, we show that GMMS -- and thus PMMS and EFX -- allocations always exist when the number of goods does not exceed the number of agents by more than two