For the problem of maximizing a monotone, submodular function with respect to a cardinality constraint $k$ on a ground set of size $n$, we provide an algorithm that achieves the state-of-the-art in both its empirical performance and its theoretical properties, in terms of adaptive complexity, query complexity, and approximation ratio; that is, it obtains, with high probability, query complexity of $O(n)$ in expectation, adaptivity of $O(\log(n))$, and approximation ratio of nearly $1-1/e$. The main algorithm is assembled from two components which may be of independent interest. The first component of our algorithm, LINEARSEQ, is useful as a preprocessing algorithm to improve the query complexity of many algorithms. Moreover, a variant of LINEARSEQ is shown to have adaptive complexity of $O( \log (n / k) )$ which is smaller than that of any previous algorithm in the literature. The second component is a parallelizable thresholding procedure THRESHOLDSEQ for adding elements with gain above a constant threshold. Finally, we demonstrate that our main algorithm empirically outperforms, in terms of runtime, adaptive rounds, total queries, and objective values, the previous state-of-the-art algorithm FAST in a comprehensive evaluation with six submodular objective functions.

翻译：针对在规模为$n$的基础集上关于基数约束$k$的单调子模函数最大化问题，我们提出了一种在经验性能与理论特性（包括自适应复杂度、查询复杂度和近似比）方面均达到当前最优水平的算法；该算法以高概率获得期望$O(n)$的查询复杂度、$O(\log(n))$的自适应复杂度以及接近$1-1/e$的近似比。该核心算法由两个具有独立价值的组件构成。第一组件LINEARSEQ可作为预处理算法显著提升多种算法的查询复杂度。此外，LINEARSEQ的变体被证明具有$O( \log (n / k) )$的自适应复杂度，低于文献中已有算法的对应值。第二组件是可并行化的阈值处理程序THRESHOLDSEQ，用于添加增益高于恒定阈值的元素。最后，通过对六种子模目标函数的综合评估，我们证明本算法在运行时间、自适应轮数、总查询量和目标函数值方面均优于先前最优算法FAST。