Optimality of Non-Adaptive Algorithms in Online Submodular Welfare Maximization with Stochastic Outcomes

2024-03-26

AI Summary

This paper studies online submodular welfare maximization under stochastic arrival and stochastic outcomes, focusing on the competitive ratio limits of non-adaptive algorithms. For objective functions satisfying only weak submodularity, we establish— for the first time—that, under polynomial-time constraints, the non-adaptive greedy algorithm achieves the optimal competitive ratio; adaptivity provides no competitive advantage in this stochastic setting. We introduce a general “deterministic-to-stochastic lifting” technique that (i) transfers deterministic upper bounds to the stochastic regime and (ii) conversely derives new tight deterministic upper bounds. Our framework unifies and strengthens existing bounds across multiple arrival models—including random-order and Poisson arrivals—and, in certain settings, strictly improves upon the performance of classical greedy algorithms.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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. 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.

Problem

• Non-adaptive algorithms in stochastic submodular welfare maximization
• Greedy algorithm achieves optimal competitive ratio
• Adaptivity offers no advantage in competitive ratio

Innovation

• Non-adaptive Greedy algorithm
• Submodular order property
• Competitive ratio optimization