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：次模最大化在人工智能领域的多个方向中得到了广泛应用，包括但不限于机器学习、计算机视觉和自然语言处理。随着这些领域中数据集规模的不断增长，开发高效且可并行的次模最大化算法变得日益迫切。衡量次模最大化算法可并行性的一个指标是其自适应复杂度，它表示可以并行执行目标函数多项式次查询的顺序轮数。本文研究了背包约束下的非单调次模最大化问题，并提出了首个组合算法，在 $\mathcal{O}(\log n)$ 的自适应复杂度下达到 $(8+\epsilon)$ 近似比，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次模最大化问题，提出了首个同时具有可证明近似比和次线性自适应复杂度的算法。作为副产品，我们证明了我们的两种算法也可应用于基数约束下的次模最大化这一特例，并且达到了与最先进算法相当的性能界限。最后，通过在现实应用中的大量实验验证了我们方法的有效性。