We consider the following Stochastic Boolean Function Evaluation problem, which is closely related to several problems from the literature. A matroid $\mathcal{M}$ (in compact representation) on ground set $E$ is given, and each element $i\in E$ is active independently with known probability $p_i\in(0,1)$. The elements can be queried, upon which it is revealed whether the respective element is active or not. The goal is to find an adaptive querying strategy for determining whether there is a basis of $\mathcal{M}$ in which all elements are active, with the objective of minimizing the expected number of queries. When $\mathcal{M}$ is a uniform matroid, this is the problem of evaluating a $k$-of-$n$ function, first studied in the 1970s. This problem is well-understood, and has an optimal adaptive strategy that can be computed in polynomial time. Taking $\mathcal{M}$ to instead be a partition matroid, we show that previous approaches fail to give a constant-factor approximation. Our main result is a polynomial-time constant-factor approximation algorithm producing a randomized strategy for this partition matroid problem. We obtain this result by combining a new technique with several well-established techniques. Our algorithm adaptively interleaves solutions to several instances of a novel type of stochastic querying problem, with a constraint on the $\textit{expected}$ cost. We believe that this type of problem is of independent interest, will spark follow-up work, and has the potential for additional applications.

翻译：我们研究以下随机布尔函数求值问题，该问题与文献中的若干问题密切相关。给定紧凑表示下的拟阵$\mathcal{M}$，其基础集为$E$，其中每个元素$i\in E$以已知概率$p_i\in(0,1)$独立激活。可通过查询元素来揭示其是否处于激活状态。目标是寻找一种自适应查询策略，以确定$\mathcal{M}$中是否存在所有元素均激活的基，且期望查询次数最小化。当$\mathcal{M}$为均匀拟阵时，该问题即$k$-of-$n$函数求值问题，最早于20世纪70年代被研究。该问题已得到充分理解，存在可在多项式时间内计算的最优自适应策略。当$\mathcal{M}$为划分拟阵时，我们证明现有方法无法给出常数因子近似解。我们的主要成果是针对该划分拟阵问题提出一种多项式时间的常数因子近似算法，该算法可生成随机化策略。该结果通过将新技术与多种成熟技术相结合而实现。我们的算法自适应地交错求解多个新型随机查询问题的实例，并对$\textit{期望}$成本施加约束。我们认为此类问题具有独立研究价值，将引发后续研究，并具备潜在的应用前景。