We study the measure of order-competitive ratio introduced by Ezra et al. [2023] for online algorithms in Bayesian combinatorial settings. In our setting, a decision-maker observes a sequence of elements that are associated with stochastic rewards that are drawn from known priors, but revealed one by one in an online fashion. The decision-maker needs to decide upon the arrival of each element whether to select it or discard it (according to some feasibility constraint), and receives the associated rewards of the selected elements. The order-competitive ratio is defined as the worst-case ratio (over all distribution sequences) between the performance of the best order-unaware and order-aware algorithms, and quantifies the loss incurred due to the lack of knowledge of the arrival order. Ezra et al. [2023] showed how to design algorithms that achieve better approximations with respect to the new benchmark (order-competitive ratio) in the single-choice setting, which raises the natural question of whether the same can be achieved in combinatorial settings. In particular, whether it is possible to achieve a constant approximation with respect to the best online algorithm for downward-closed feasibility constraints, whether $\omega(1/n)$-approximation is achievable for general (non-downward-closed) feasibility constraints, or whether a convergence rate to $1$ of $o(1/\sqrt{k})$ is achievable for the multi-unit setting. We show, by devising novel constructions that may be of independent interest, that for all three scenarios, the asymptotic lower bounds with respect to the old benchmark, also hold with respect to the new benchmark.

翻译：我们研究了Ezra等人[2023]针对贝叶斯组合设定下在线算法提出的顺序竞争比度量指标。在该设定中，决策者观察一系列与随机奖励相关的元素，这些奖励由已知先验分布生成，但需以在线方式逐个揭示。决策者需根据可行性约束条件，在每项元素到达时决定是否选择或丢弃，并获得所选元素的对应奖励。顺序竞争比被定义为最佳无视顺序算法与知晓顺序算法性能之间最坏情况比值（针对所有分布序列），量化了因缺乏到达顺序知识而产生的性能损失。Ezra等人[2023]展示了如何在单选择设定中设计算法以在新基准（顺序竞争比）下实现更优近似，这自然引发了一个问题：在组合设定中是否也能实现同样效果。具体而言：针对下闭可行性约束，是否可能实现相对于最优在线算法的常数近似？针对一般（非下闭）可行性约束，能否实现$\omega(1/n)$-近似？针对多单元设定，是否能达到收敛到$1$的$o(1/\sqrt{k})$收敛速率？通过设计具有独立意义的新型构造，我们证明在三种场景下，针对旧基准的渐近下界同样适用于新基准。