Consider the triplet $(E, \mathcal{P}, \pi)$, where $E$ is a finite ground set, $\mathcal{P} \subseteq 2^E$ is a collection of subsets of $E$ and $\pi : \mathcal{P} \rightarrow [0,1]$ is a requirement function. Given a vector of marginals $\rho \in [0, 1]^E$, our goal is to find a distribution for a random subset $S \subseteq E$ such that $\operatorname{Pr}[e \in S] = \rho_e$ for all $e \in E$ and $\operatorname{Pr}[P \cap S \neq \emptyset] \geq \pi_P$ for all $P \in \mathcal{P}$, or to determine that no such distribution exists. Generalizing results of Dahan, Amin, and Jaillet, we devise a generic decomposition algorithm that solves the above problem when provided with a suitable sequence of admissible support candidates (ASCs). We show how to construct such ASCs for numerous settings, including supermodular requirements, Hoffman-Schwartz-type lattice polyhedra, and abstract networks where $\pi$ fulfils a conservation law. The resulting algorithm can be carried out efficiently when $\mathcal{P}$ and $\pi$ can be accessed via appropriate oracles. For any system allowing the construction of ASCs, our results imply a simple polyhedral description of the set of marginal vectors for which the decomposition problem is feasible. Finally, we characterize balanced hypergraphs as the systems $(E, \mathcal{P})$ that allow the perfect decomposition of any marginal vector $\rho \in [0,1]^E$, i.e., where we can always find a distribution reaching the highest attainable probability $\operatorname{Pr}[P \cap S \neq \emptyset] = \min \{ \sum_{e \in P} \rho_e, 1\}$ for all $P \in \mathcal{P}$.

翻译：考虑三元组 $(E, \mathcal{P}, \pi)$，其中 $E$ 是有限基集，$\mathcal{P} \subseteq 2^E$ 是 $E$ 的子集族，$\pi : \mathcal{P} \rightarrow [0,1]$ 是要求函数。给定边际向量 $\rho \in [0, 1]^E$，我们的目标是找到一个随机子集 $S \subseteq E$ 的分布，使得对所有 $e \in E$ 有 $\operatorname{Pr}[e \in S] = \rho_e$，且对所有 $P \in \mathcal{P}$ 有 $\operatorname{Pr}[P \cap S \neq \emptyset] \geq \pi_P$，或判定不存在这样的分布。推广 Dahan、Amin 和 Jaillet 的结果，我们设计了一种通用的分解算法，当提供合适的可接受支撑候选序列（ASCs）时，该算法能解决上述问题。我们展示了如何在多种场景下构造此类 ASC，包括超模要求、Hoffman-Schwartz 型格点多面体以及满足守恒律的抽象网络。当 $\mathcal{P}$ 和 $\pi$ 可通过适当预言机访问时，所得算法可高效执行。对于任何允许构造 ASC 的系统，我们的结果给出了边际向量集合（分解问题可行）的简单多面体描述。最后，我们将平衡超图刻画为系统 $(E, \mathcal{P})$，该类系统允许任意边际向量 $\rho \in [0,1]^E$ 的完美分解，即总能找到一个分布使得对所有 $P \in \mathcal{P}$ 达到最高可达概率 $\operatorname{Pr}[P \cap S \neq \emptyset] = \min \{ \sum_{e \in P} \rho_e, 1\}$。