In this paper, we study the non-monotone adaptive submodular maximization problem subject to a cardinality constraint. We first revisit the adaptive random greedy algorithm proposed in [13], where they show that this algorithm achieves a $1/e$ approximation ratio if the objective function is adaptive submodular and pointwise submodular. It is not clear whether the same guarantee holds under adaptive submodularity (without resorting to pointwise submodularity) or not. Our first contribution is to show that the adaptive random greedy algorithm achieves a $1/e$ approximation ratio under adaptive submodularity. One limitation of the adaptive random greedy algorithm is that it requires $O(n\times k)$ value oracle queries, where n is the size of the ground set and k is the cardinality constraint. Our second contribution is to develop the first linear-time algorithm for the non-monotone adaptive submodular maximization problem. Our algorithm achieves a $1/e-ϵ$ approximation ratio (this bound is improved to $1-1/e-ϵ$ for monotone case), using only $O(n{ϵ}^{-2}\log {ϵ}^{-1})$ value oracle queries. Notably, $O(n{ϵ}^{-2}\log {ϵ}^{-1})$ is independent of the cardinality constraint. For the monotone case, we propose a faster algorithm that achieves a $1-1/e-ϵ$ approximation ratio in expectation with $O(n\log \frac{1}{ϵ})$ value oracle queries. We also generalize our study by considering a partition matroid constraint, and develop a linear-time algorithm for monotone and fully adaptive submodular functions.

在本文中，我们研究了受基数约束的非单调自适应次模最大化问题。我们首先回顾[13]中提出的自适应随机贪婪算法，他们表明该算法实现了$\mathrm{1个}/Ë$如果目标函数是自适应子模和点式子模，则为近似值。尚不清楚在自适应子模态下（不求助于逐点子模态下）是否存在相同的保证。我们的第一个贡献是证明自适应随机贪婪算法实现了$\mathrm{1个}/Ë$自适应亚模态下的近似比。自适应随机贪婪算法的局限性在于它需要$Ø（ñ\times ķ）$值oracle查询，其中n是地面集的大小，k是基数约束。我们的第二个贡献是为非单调自适应亚模最大化问题开发了第一个线性时间算法。我们的算法实现了$\mathrm{1个}/Ë-ϵ$ 近似比率（此界限已改进为 $\mathrm{1个}-\mathrm{1个}/Ë-ϵ$ （对于单调情况），仅使用 $Ø（ñ{ϵ}^{-2}\mathrm{日志}{ϵ}^{-\mathrm{1个}}）$值oracle查询。值得注意的是$Ø（ñ{ϵ}^{-2}\mathrm{日志}{ϵ}^{-\mathrm{1个}}）$与基数约束无关。对于单调情况，我们提出了一种更快的算法，可以实现$\mathrm{1个}-\mathrm{1个}/Ë-ϵ$ 预期中的近似比 $Ø（ñ\mathrm{日志}\frac{\mathrm{1个}}{ϵ}）$值oracle查询。我们还通过考虑分区拟阵约束来概括我们的研究，并为单调和完全自适应的子模函数开发线性时间算法。