On the Complexity of Maximizing Social Welfare within Fair Allocations of Indivisible Goods

We consider the classical fair division problem which studies how to allocate resources fairly and efficiently. We give a complete landscape on the computational complexity and approximability of maximizing the social welfare within (1) envy-free up to any item (EFX) and (2) envy-free up to one item (EF1) allocations of indivisible goods for both normalized and unnormalized valuations. We show that a partial EFX allocation may have a higher social welfare than a complete EFX allocation, while it is well-known that this is not true for EF1 allocations. Thus, our first group of results focuses on the problem of maximizing social welfare subject to (partial) EFX allocations. For $n=2$ agents, we provide a polynomial time approximation scheme (PTAS) and an NP-hardness result. For a general number of agents $n>2$, we present algorithms that achieve approximation ratios of $O(n)$ and $O(\sqrt{n})$ for unnormalized and normalized valuations, respectively. These results are complemented by the asymptotically tight inapproximability results. We also study the same constrained optimization problem for EF1. For $n=2$, we show a fully polynomial time approximation scheme (FPTAS) and complement this positive result with an NP-hardness result. For general $n$, we present polynomial inapproximability ratios for both normalized and unnormalized valuations. Our results also imply the price of EFX is $\Theta(\sqrt{n})$ for normalized valuations, which is unknown in the previous literature.