In traditional resource/task allocation models, an agent's value on an item is fixed regardless of when, where, or how the agent consumes the item. In this work, we introduce a more general framework involving a set of positions (for example, roles within a company), where agents have valuations that depend on the specific positions they are assigned. This setting is well motivated as the agent faces different resources and support at different positions, and thus may need varying levels of effort to complete the item, supposing the items are chores. We further consider the constrained setting when each item can only be allocated to a certain set of positions (e.g., a research project cannot be assigned to an administrative staff). We particularly consider the case when the constraints form a graph where edges are items and vertices are positions, so an item can only be allocated to an incident position, which is known as the orientation problem [Christodoulou et al. EC 2023]. In this paper, fairness is measured by maximin share (MMS), and efficiency is measured by Pareto and social optimality. We present a complete set of results on the computational complexity and approximation algorithms for computing efficient and fair allocations.

We adopt a parametric approach to analyze the worst-case degradation in social welfare when the allocation of indivisible goods is constrained to be fair. Specifically, we are concerned with cardinality-constrained allocations, which require that each agent has at most k items in their allocated bundle. We propose the notion of the price of cardinality, which captures the worst-case multiplicative loss of utilitarian or egalitarian social welfare resulting from imposing the cardinality constraint. We then characterize tight or almost-tight bounds on the price of cardinality as exact functions of the instance parameters, demonstrating how the social welfare improves as k is increased. In particular, one of our main results refines and generalizes the existing asymptotic bound of Θ(√n) on the price of balancedness. We also further extend our analysis to the problem where the items are partitioned into disjoint categories, and each category has its own cardinality constraint. Through a parametric study of the price of cardinality, we provide a framework which aids decision makers in choosing an ideal level of cardinality-based fairness, using their knowledge of the potential loss of utilitarian and egalitarian social welfare.

Envy-freeness is one of the most important fairness concerns when allocating items. We study envy-free house allocation when agents have uncertain preferences over items and consider several well-studied preference uncertainty models. The central problem that we focus on is computing an allocation that has the highest probability of being envy-free. We show that each model leads to a distinct set of algorithmic and complexity results, including detailed results on (in-)approximability. En route, we consider two related problems of checking whether there exists an allocation that is possibly or necessarily envy-free. We give a complete picture of the computational complexity of these two problems for all the uncertainty models we consider.

Discrete cake cutting is a fundamental model in fair resource allocation where the indivisible resources are located on a path. It is well motivated that, in reality, each agent is interested in receiving a contiguous block of items. An important question therein is to understand the economic efficiency loss by restricting the allocations to be fair, which is quantified as price of fairness (PoF). Informally, PoF is the worst-case ratio between the unconstrained optimal welfare and the optimal welfare achieved by fair allocations. Suksompong [Discret. Appl. Math., 2019] has studied this problem, where fairness is measured by the ideal criteria such as proportionality (PROP). A PROP allocation, however, may not exist in discrete cake cutting settings. Therefore, in this work, we revisit this problem and focus on the relaxed notions whose existence is guaranteed. We study both utilitarian and egalitarian welfare, and our results show significant differences between the PoF of guaranteed fairness notions and that of the ideal notions.

Abstract We study fair allocations of indivisible goods and chores in conjunction with system efficiency, measured by two social welfare functions, namely utilitarian and egalitarian welfare. To model preference, each agent is associated with a cardinal and additive valuation function. The fairness criteria we are concerned with are equitability up to any item (EQX) and equitability up to one item (EQ1). For the trade-off between fairness and efficiency, we investigate efficiency loss under these fairness constraints and establish the price of fairness. From the computational perspective, we provide a complete picture of the computational complexity of (i) deciding the existence of an EQX/EQ1 and welfare-maximizing allocation; (ii) computing a welfare maximizer among all EQX/EQ1 allocations.

Abstract We study several fairness notions in allocating indivisible chores (i.e., items with disutilities) to agents who have additive and submodular cost functions. The fairness criteria we are concerned with are envy-free up to any item, envy-free up to one item, maximin share (MMS), and pairwise maximin share (PMMS), which are proposed as relaxations of envy-freeness in the setting of additive cost functions. For allocations under each fairness criterion, we establish their approximation guarantee for other fairness criteria. Under the additive setting, our results show strong connections between these fairness criteria and, at the same time, reveal intrinsic differences between goods allocation and chores allocation. However, such strong relationships cannot be inherited by the submodular setting, under which PMMS and MMS are no longer relaxations of envy-freeness and, even worse, few non-trivial guarantees exist. We also investigate efficiency loss under these fairness constraints and establish their prices of fairness.

We incorporate group fairness into the algorithmic centroid clustering problem, where $k$ centers are to be located to serve $n$ agents distributed in a metric space. We refine the notion of proportional fairness proposed in [Chen et al., ICML 2019] as {\em core fairness}. A $k$-clustering is in the core if no coalition containing at least $n/k$ agents can strictly decrease their total distance by deviating to a new center together. Our solution concept is motivated by the situation where agents are able to coordinate and utilities are transferable. A string of existence, hardness and approximability results is provided. Particularly, we propose two dimensions to relax core requirements: one is on the degree of distance improvement, and the other is on the size of deviating coalition. For both relaxations and their combination, we study the extent to which relaxed core fairness can be satisfied in metric spaces including line, tree and general metric space, and design approximation algorithms accordingly. We also conduct experiments on synthetic and real-world data to examine the performance of our algorithms.