We generalize the problem of online submodular welfare maximization to incorporate various stochastic elements that have gained significant attention in recent years. We show that a non-adaptive Greedy algorithm, which is oblivious to the realization of these stochastic elements, achieves the best possible competitive ratio among all polynomial-time algorithms, including adaptive ones, unless NP$=$RP. This result holds even when the objective function is not submodular but instead satisfies the weaker submodular order property. Our results unify and strengthen existing competitive ratio bounds across well-studied settings and diverse arrival models, showing that, in general, adaptivity to stochastic elements offers no advantage in terms of competitive ratio. To establish these results, we introduce a technique that lifts known results from the deterministic setting to the generalized stochastic setting. The technique has broad applicability, enabling us to show that, in certain special cases, non-adaptive Greedy-like algorithms outperform the Greedy algorithm and achieve the optimal competitive ratio. We also apply the technique in reverse to derive new upper bounds on the performance of Greedy-like algorithms in deterministic settings by leveraging upper bounds on the performance of non-adaptive algorithms in stochastic settings.

翻译：我们将在线次模福利最大化问题推广至包含近年来备受关注的各种随机元素。我们证明，一个非自适应的贪心算法（该算法对这些随机元素的实现是未知的）在所有多项式时间算法（包括自适应算法）中实现了最佳可能的竞争比，除非NP$=$RP。这一结果甚至在目标函数不是次模而是满足较弱的次模序性质时依然成立。我们的结果统一并强化了现有在多种已深入研究的设定和不同到达模型下的竞争比界限，表明一般而言，对随机元素的自适应性在竞争比方面不提供任何优势。为确立这些结果，我们引入了一种技术，将确定性设定下的已知结果提升至广义随机设定。该技术具有广泛的适用性，使我们能够证明，在某些特殊情况下，类贪心非自适应算法优于贪心算法并达到最优竞争比。我们还反向应用该技术，通过利用随机设定下非自适应算法的性能上界，推导出确定性设定下类贪心算法性能的新上界。