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 requesting a set of items, each with a value drawn from a known distribution. We study the efficiency of static and anonymous bundle pricing in environments where the buyers' valuations exhibit strong complementarities. In such settings, standard item pricing fails to leverage item multiplicities, while static bundle pricing mechanisms are only known for very restricted domains and their analysis relies on domain-specific arguments. We develop a unified bundle pricing framework for online resource allocation in three well-studied domains with complementarities: (i) single-minded combinatorial auctions with maximum bundle size $d$; (ii) general single-minded combinatorial auctions; and (iii) network routing, where each buyer aims to route a unit of flow from a source node $s$ to a target node $t$ in a capacitated network. Our approach yields static and anonymous bundle pricing mechanisms whose performance improves exponentially with item multiplicity. For the $d$-single-minded setting with minimum item multiplicity $B$, we obtain an $O(d^{1/B})$-competitive mechanism. For general single-minded combinatorial auctions and online network routing, 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 $ \widetildeΩ(m^{1/(B+2)})$ for single-minded combinatorial auctions and $ \widetildeΩ(d^{1/(B+1)})$ for the $d$-single-minded setting. Our constructions exploit 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，如何将有限资源高效分配给买家的问题。在我们的设定中，买家按顺序到达，请求一组物品，每个物品的价值服从已知分布。我们研究了在买家估值表现出强互补性的环境中，静态匿名捆绑定价的效率。在此类场景中，标准物品定价无法利用物品的重复性，而静态捆绑定价机制仅适用于高度受限的领域，且其分析依赖于特定领域的论证。我们针对三个具有互补性的经典领域，开发了一个统一的在线资源分配捆绑定价框架：(i) 最大捆绑规模为 $d$ 的单一需求组合拍卖；(ii) 一般单一需求组合拍卖；以及 (iii) 网络路由，其中每个买家旨在从源节点 $s$ 到目标节点 $t$ 在容量受限的网络中路由一个单位的流量。我们的方法产生了静态匿名捆绑定价机制，其性能随物品重复性呈指数级提升。对于最小物品重复性为 $B$ 的 $d$-单一需求设定，我们获得了一个 $O(d^{1/B})$-竞争性机制。对于一般单一需求组合拍卖和在线网络路由，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对这些结果进行了补充。我们证明，对于单一需求组合拍卖，任何在线算法都无法实现优于 $ \widetildeΩ(m^{1/(B+2)})$ 的竞争比；对于 $d$-单一需求设定，无法实现优于 $ \widetildeΩ(d^{1/(B+1)})$ 的竞争比。我们的构建利用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层联系。