On the Complexity of Maximizing Social Welfare within Fair Allocations of Indivisible Goods
arxiv(2023)
摘要
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.
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