Approximation Algorithm for Computing Budget-Feasible EF1 Allocations

We study algorithmic fairness in a budget-feasible resource allocation problem. In this problem, a set of items with varied sizes and values are to be allocated to a group of agents, while each agent has a budget constraint on the total size of items she can receive. An envy-free (EF) allocation is defined in this context as one in which no agent envies another for the items they get and, in addition, no agent envies the charity, who is automatically endowed with all the unallocated items. Since EF allocations barely exist even without budget constraints, we are interested in the relaxed notion of envy-freeness up to one item (EF1). In this paper, we further the recent progress towards understanding the existence and approximations of EF1 (or EF2) allocations. We propose a polynomial-time algorithm that computes a 1/2-approximate EF1 allocation for an arbitrary number of agents with heterogeneous budgets. For the uniform-budget and two-agent cases, we present a polynomial-time algorithm that computes an exact EF1 allocation. We also consider the large budget setting, where the item sizes are infinitesimal relative to the agents' budgets. We show that both the allocations that maximize the Nash social welfare and the allocations that our main algorithm computes are EF1 in the limit.