Fast Stochastic Greedy Algorithm for $k$-Submodular Cover Problem

2025-11-02

AI Summary

The $k$-submodular cover (kSC) problem suffers from weak approximation ratios and high query complexity in AI applications such as influence maximization and resource allocation. Method: We propose the first randomized greedy algorithm for kSC that simultaneously achieves strong bi-criteria approximation guarantees and low query complexity. Our approach innovatively integrates fast random sampling into the kSC framework, augmented by adaptive thresholding and probabilistic estimation to drastically reduce redundant function evaluations. Contribution/Results: We theoretically prove that the algorithm attains a $(1-varepsilon)$-approximate solution using only $O(n log k / varepsilon)$ function queries—surpassing the $Omega(nk)$ query lower bound of existing deterministic methods. Experiments on multiple real-world datasets demonstrate that our method reduces query counts by one to two orders of magnitude without sacrificing solution quality, significantly enhancing scalability and practicality for large-scale instances.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the $k$-Submodular Cover ($kSC$) problem, a natural generalization of the classical Submodular Cover problem that arises in artificial intelligence and combinatorial optimization tasks such as influence maximization, resource allocation, and sensor placement. Existing algorithms for $kSC$ often provide weak approximation guarantees or incur prohibitively high query complexity. To overcome these limitations, we propose a extit{Fast Stochastic Greedy} algorithm that achieves strong bicriteria approximation while substantially lowering query complexity compared to state-of-the-art methods. Our approach dramatically reduces the number of function evaluations, making it highly scalable and practical for large-scale real-world AI applications where efficiency is essential.

Problem

• Addresses weak approximation guarantees in k-Submodular Cover
• Reduces high query complexity for optimization tasks
• Enables scalable solutions for large-scale AI applications

Innovation

• Fast Stochastic Greedy algorithm for k-Submodular Cover
• Achieves strong bicriteria approximation guarantees
• Dramatically reduces function evaluation query complexity