Fair Division of Indivisible Goods for a Class of Concave Valuations

We study the fair and efficient allocation of a set of indivisible goods among agents, where each good has several copies, and each agent has an additively separable concave valuation function with a threshold. These valuations capture the property of diminishing marginal returns, and they are more general than the well-studied case of additive valuations. We present a polynomial-time algorithm that approximates the optimal Nash social welfare (NSW) up to a factor of e1/e ≈ 1.445. This matches with the state-of-the-art approximation factor for additive valuations. The computed allocation also satisfies the popular fairness guarantee of envy-freeness up to one good (EF1) up to a factor of 2 + ε. For instances without thresholds, it is also approximately Pareto-optimal. For instances satisfying a large market property, we show an improved approximation factor. Lastly, we show that the upper bounds on the optimal NSW introduced in Cole and Gkatzelis (2018) and Barman et al. (2018) have the same value.