This work proposes an efficient parallel algorithm for non-monotone submodular maximization under a knapsack constraint problem over the ground set of size $n$. Our algorithm improves the best approximation factor of the existing parallel one from $8+\epsilon$ to $7+\epsilon$ with $O(\log n)$ adaptive complexity. The key idea of our approach is to create a new alternate threshold algorithmic framework. This strategy alternately constructs two disjoint candidate solutions within a constant number of sequence rounds. Then, the algorithm boosts solution quality without sacrificing the adaptive complexity. Extensive experimental studies on three applications, Revenue Maximization, Image Summarization, and Maximum Weighted Cut, show that our algorithm not only significantly increases solution quality but also requires comparative adaptivity to state-of-the-art algorithms.

本研究针对在背包约束下非单调子模极大化问题，提出了一种高效的并行算法，有效将现有并行算法的最佳近似因子从$8+\epsilon$提高到$7+\epsilon$，且具备$O(\log n)$的自适应复杂度。通过构建新的交替阈值算法框架，该算法在保证自适应复杂度的同时显著提升了解的质量，在收入最大化、图像摘要和最大加权切割等多个应用上进行了广泛的实验研究，展现出优越的性能。 

非单调子模极大化的改进并行算法及其背包约束