Online Resource Allocation addresses the problem of efficiently allocating limited resources to buyers with incomplete knowledge of future requests. In our setting, buyers arrive sequentially demanding a set of items, each with a value drawn from a known distribution. We study environments where buyers' valuations exhibit complementarities. In such settings, standard item-pricing mechanisms fail to leverage item multiplicities, while existing static bundle-pricing mechanisms rely on problem-specific arguments that do not generalize. We develop a unified technique for online resource allocation with complementarities for three domains: (i) single-minded combinatorial auctions with maximum bundle size $d$, (ii) general single-minded combinatorial auctions, and (iii) a graph-based routing model in which buyers request to route a unit of flow from a source node $s$ to a target node $t$ in a capacitated graph. Our approach yields static and anonymous bundle-pricing mechanisms whose performance improves exponentially with item capacities. For the $d$-single-minded setting with minimum item capacity $B$, we obtain an $O(d^{1/B})$-competitive mechanism, recovering the known $O(d)$ bound for unit capacities ($B=1$) and achieving exponentially better guarantees as capacities grow. For general single-minded combinatorial auctions and the graph-routing model, we obtain $O(m^{1/(B+1)})$-competitive mechanisms, where $m$ is the number of items. We complement these results with information-theoretic lower bounds. We show that no online algorithm can achieve a competitive ratio better than $Ω((m/\ln m)^{1/(B+2)})$ in the general single-minded setting and $Ω((d/\ln d)^{1/(B+1)})$ in the $d$-single-minded setting. In doing so, we reveal a deep connection to the extremal combinatorics problem of determining the maximum number of qualitatively independent partitions of a ground set.

翻译：在线资源分配旨在解决在未知未来请求的情况下，将有限资源高效分配给买家的问题。在我们的设定中，买家按序到达，要求一组物品，每个物品的价值服从已知分布。我们研究买家估值呈现互补性的环境。在此类场景中，标准物品定价机制无法利用物品的多重性，而现有静态捆绑定价机制依赖于无法泛化的特定问题论证。我们针对三个领域开发了一种统一的在线资源分配技术以处理互补性：(i) 最大捆绑规模为 $d$ 的单向组合拍卖，(ii) 通用单向组合拍卖，以及 (iii) 基于图的路由模型，其中买家请求在容量图中从源节点 $s$ 到目标节点 $t$ 路由一个单位流量。我们的方法产生了静态且匿名的捆绑定价机制，其性能随物品容量呈指数级提升。对于最小物品容量为 $B$ 的 $d$-单向设定，我们获得了 $O(d^{1/B})$-竞争性机制，恢复了单位容量 ($B=1$) 时已知的 $O(d)$ 界限，并在容量增长时实现了指数级更优的保证。对于通用单向组合拍卖和图路由模型，我们获得了 $O(m^{1/(B+1)})$-竞争性机制，其中 $m$ 为物品数量。我们通过信息论下界补充了这些结果。我们证明，在通用单向设定中，任何在线算法都无法达到优于 $Ω((m/\ln m)^{1/(B+2)})$ 的竞争比；在 $d$-单向设定中，无法达到优于 $Ω((d/\ln d)^{1/(B+1)})$ 的竞争比。在此过程中，我们揭示了与确定基础集上定性独立划分最大数量的极值组合问题之间的深刻联系。