Fairly Dividing Non-identical Random Items: Just Sample or Match

We study the question of existence and fast computation of fair and efficient allocations of indivisible resources among agents with additive valuations. As such allocations may not exist for arbitrary instances, we ask if they exist for \textit{typical} or \textit{random} instances, meaning when the utility values of agents for the resources are drawn from certain distributions. If such allocations exist with high probability for typical instances, and furthermore if they can be computed efficiently, this would imply that we could quickly resolve a real world resource allocation scenario in a fair and efficient manner with high probability. This implication has made this setting popular and well studied in fair resource allocation. In this paper, we extend the previously studied formal models of this problem to non-identical items. We assume that every item is associated with a distribution $\mathcal{U}_j$, and every agent's utility value for the item is drawn independently from $\mathcal{U}_j$. We show that envy-free fair and maximum social welfare efficient allocations exist with high probability in the asymptotic setting, meaning when the number of agents $n$ and items $m$ are large. Further we show that when $m=O(n\log n),$ then by only sampling $O(\log m)$ or $O((\log m)^2)$ utility values per item instead of all the $n,$ we can compute these allocations in $\tilde{O}(m)$ time. Finally, we simulate our algorithms on randomly generated instances and show that even for small instances, we suffer small multiplicative losses in the fairness and efficiency guarantees even for small sized instances, and converge to fully optimal guarantees quickly.