Submodular maximization has found extensive applications in various domains within the field of artificial intelligence, including but not limited to machine learning, computer vision, and natural language processing. With the increasing size of datasets in these domains, there is a pressing need to develop efficient and parallelizable algorithms for submodular maximization. One measure of the parallelizability of a submodular maximization algorithm is its adaptive complexity, which indicates the number of sequential rounds where a polynomial number of queries to the objective function can be executed in parallel. In this paper, we study the problem of non-monotone submodular maximization subject to a knapsack constraint, and propose the first combinatorial algorithm achieving an $(8+\epsilon)$-approximation under $\mathcal{O}(\log n)$ adaptive complexity, which is \textit{optimal} up to a factor of $\mathcal{O}(\log\log n)$. Moreover, we also propose the first algorithm with both provable approximation ratio and sublinear adaptive complexity for the problem of non-monotone submodular maximization subject to a $k$-system constraint. As a by-product, we show that our two algorithms can also be applied to the special case of submodular maximization subject to a cardinality constraint, and achieve performance bounds comparable with those of state-of-the-art algorithms. Finally, the effectiveness of our approach is demonstrated by extensive experiments on real-world applications.

翻译：子模最大化在人工智能领域的各个分支中得到了广泛应用，包括但不限于机器学习、计算机视觉和自然语言处理。随着这些领域数据规模的不断增大，开发高效且可并行的子模最大化算法变得迫切。子模最大化算法并行能力的一个衡量标准是其自适应复杂度，它表示可以并行执行对目标函数进行多项式查询的连续轮次数量。本文研究了在背包约束下的非单调子模最大化问题，并提出了首个在 $\mathcal{O}(\log n)$ 自适应复杂度下达到 $(8+\epsilon)$ 近似比的组合算法，该复杂度在 $\mathcal{O}(\log\log n)$ 因子内是\textit{最优}的。此外，我们还针对 $k$-系统约束下的非单调子模最大化问题，提出了首个同时具有可证明近似比和亚线性自适应复杂度的算法。作为副产品，我们证明了这两种算法也可应用于基数约束下的子模最大化特例，并取得了与当前最优算法相当的性能界。最后，通过在真实世界应用上的大量实验，验证了我们方法的有效性。