Almost Group Envy-free Allocation of Indivisible Goods and Chores

We consider a multi-agent resource allocation setting in which an agent's utility may decrease or increase when an item is allocated. We take the group envy-freeness concept that is well-established in the literature and present stronger and relaxed versions that are especially suitable for the allocation of indivisible items. Of particular interest is a concept called group envy-freeness up to one item (GEF1). We then present a clear taxonomy of the fairness concepts. We study which fairness concepts guarantee the existence of a fair allocation under which preference domain. For two natural classes of additive utilities, we design polynomial-time algorithms to compute a GEF1 allocation. We also prove that checking whether a given allocation satisfies GEF1 is coNP-complete when there are either only goods, only chores or both

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

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

Envy-free Relaxations for Goods, Chores, and Mixed Items

In fair division problems, we are given a set $S$ of $m$ items and a set $N$ of $n$ agents with individual preferences, and the goal is to find an allocation of items among agents so that each agent finds the allocation fair. There are several established fairness concepts and envy-freeness is one of the most extensively studied ones. However envy-free allocations do not always exist when items are indivisible and this has motivated relaxations of envy-freeness: envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) are two well-studied relaxations. We consider the problem of finding EF1 and EFX allocations for utility functions that are not necessarily monotone, and propose four possible extensions of different strength to this setting. In particular, we present a polynomial-time algorithm for finding an EF1 allocation for two agents with arbitrary utility functions. An example is given showing that EFX allocations need not exist for two agents with non-monotone, non-additive, identical utility functions. However, when all agents have monotone (not necessarily additive) identical utility functions, we prove that an EFX allocation of chores always exists. As a step toward understanding the general case, we discuss two subclasses of utility functions: Boolean utilities that are $\{0,+1\}$-valued functions, and negative Boolean utilities that are $\{0,-1\}$-valued functions. For the latter, we give a polynomial time algorithm that finds an EFX allocation when the utility functions are identical.Comment: 21 pages, 1 figur

The Good, the Bad and the Submodular: Fairly Allocating Mixed Manna Under Order-Neutral Submodular Preferences

We study the problem of fairly allocating indivisible goods (positively valued items) and chores (negatively valued items) among agents with decreasing marginal utilities over items. Our focus is on instances where all the agents have simple preferences; specifically, we assume the marginal value of an item can be either $-1$, $0$ or some positive integer $c$. Under this assumption, we present an efficient algorithm to compute leximin allocations for a broad class of valuation functions we call order-neutral submodular valuations. Order-neutral submodular valuations strictly contain the well-studied class of additive valuations but are a strict subset of the class of submodular valuations. We show that these leximin allocations are Lorenz dominating and approximately proportional. We also show that, under further restriction to additive valuations, these leximin allocations are approximately envy-free and guarantee each agent their maxmin share. We complement this algorithmic result with a lower bound showing that the problem of computing leximin allocations is NP-hard when $c$ is a rational number