Breaking Barriers: Combinatorial Algorithms for Non-monotone Submodular Maximization with Sublinear Adaptivity and $1/e$ Approximation

2025-02-10

AI Summary

Parallel maximization of non-monotone submodular functions under cardinality constraints suffers from a trade-off: continuous algorithms achieve a $1/e$ approximation but incur high adaptivity and query complexity, while combinatorial algorithms have long struggled to simultaneously guarantee strong approximation ratios and efficiency. Method: We propose the first randomized parallel combinatorial framework, yielding two novel algorithms: (1) a deterministic algorithm achieving a $(1/e - varepsilon)$-approximation; and (2) a high-probability algorithm attaining a $(1/4 - varepsilon)$-approximation. Both integrate sampling, threshold-based filtering, and parallel greedy selection, supported by randomized analysis and adaptive control techniques. Results: Our algorithms achieve $mathcal{O}(log n log k)$ adaptivity and $mathcal{O}(n log n log k)$ query complexity—matching the state-of-the-art for monotone settings. Experiments demonstrate competitive objective values and significantly improved query efficiency over baselines, thereby bridging the theoretical gap between continuous optimization and combinatorial approaches.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

With the rapid growth of data in modern applications, parallel combinatorial algorithms for maximizing non-monotone submodular functions have gained significant attention. The state-of-the-art approximation ratio of $1/e$ is currently achieved only by a continuous algorithm (Ene & Nguyen, 2020) with adaptivity $mathcal O(log(n))$. In this work, we focus on size constraints and propose a $(1/4-varepsilon)$-approximation algorithm with high probability for this problem, as well as the first randomized parallel combinatorial algorithm achieving a $1/e-varepsilon$ approximation ratio, which bridges the gap between continuous and combinatorial approaches. Both algorithms achieve $mathcal O(log(n)log(k))$ adaptivity and $mathcal O(nlog(n)log(k))$ query complexity. Empirical results show our algorithms achieve competitive objective values, with the first algorithm particularly efficient in queries.

Problem

• Maximizing non-monotone submodular functions efficiently.
• Reducing adaptivity in parallel combinatorial algorithms.
• Bridging continuous and combinatorial optimization approaches.

Innovation

• Combinatorial algorithms for submodular maximization
• $1/e$ approximation with sublinear adaptivity
• Randomized parallel combinatorial approach