Approximability Landscape of Welfare Maximization within Fair Allocations

Fair allocation of indivisible goods studies allocating $m$ goods among $n$ agents in a fair manner. While fairness is a fundamental requirement in many real-world applications, it often conflicts with (economic) efficiency. This raises a natural and important question: How can we identify the most welfare-efficient allocation among all fair allocations? This paper answers from the perspective of computational complexity. Specifically, we study the problem of maximizing utilitarian social welfare under two widely studied fairness criteria: envy-freeness up to any item (EFX) and envy-freeness up to one item (EF1). We examine both normalized and unnormalized valuations, where normalized valuations require that each agent's total utility for all items is identical. The key contributions of this paper can be summarized as follows: (i) we sketch the complete complexity landscape of welfare maximization subject to fair allocation constraints; and (ii) we provide interesting bounds on the price of fairness for both EFX and EF1. Specifically: (1) For $n=2$ agents, we develop polynomial-time approximation schemes (PTAS) and provide NP-hardness results for EFX and EF1 constraints; (2) For $n>2$ agents, under EFX constraints, we design algorithms that achieve approximation ratios of $O(n)$ and $O(\sqrt{n})$ for unnormalized and normalized valuations, respectively. These results are complemented by asymptotically tight inapproximability results. We also obtain similar results for EF1 constraints; (3) When the number of agents is a fixed constant, we show that the optimal solution can be computed in polynomial time by slightly relaxing the fairness constraints, whereas exact fairness leads to strong inapproximability; (4) Furthermore, our results imply the price of EFX is $\Theta(\sqrt{n})$ for normalized valuations, which is unknown in the literature.