Given a set system $(E, \mathcal{P})$ with $\rho \in [0, 1]^E$ and $\pi \in [0,1]^{ \mathcal{P}}$, our goal is to find a probability distribution for a random set $S \subseteq E$ 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 $P \in \mathcal{P}$. We extend the results of Dahan, Amin, and Jaillet (MOR 2022) who studied this problem motivated by a security game in a directed acyclic graph (DAG). We focus on the setting where $\pi$ is of the affine form $\pi_P = 1 - \sum_{e \in P} \mu_e$ for $\mu \in [0, 1]^E$. A necessary condition for the existence of the desired distribution is that $\sum_{e \in P} \rho_e \geq \pi_P$ for all $P \in \mathcal{P}$. We show that this condition is sufficient if and only if $\mathcal{P}$ has the weak max-flow/min-cut property. We further provide an efficient combinatorial algorithm for computing the corresponding distribution in the special case where $(E, \mathcal{P})$ is an abstract network. As a consequence, equilibria for the security game by Dahan et al. can be efficiently computed in a wide variety of settings (including arbitrary digraphs). As a subroutine of our algorithm, we provide a combinatorial algorithm for computing shortest paths in abstract networks, partially answering an open question by McCormick (SODA 1996). We further show that a conservation law proposed by Dahan et al. for the requirement vector $\pi$ in DAGs can be reduced to the setting of affine requirements described above.

翻译：给定集合系统 $(E, \mathcal{P})$，其中 $\rho \in [0, 1]^E$ 和 $\pi \in [0,1]^{ \mathcal{P}}$，我们的目标是找到一个随机子集 $S \subseteq E$ 的概率分布，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 $\operatorname{Pr}[P \cap S \neq \emptyset] \geq \pi_P$。我们扩展了 Dahan、Amin 和 Jaillet (MOR 2022) 的研究结果，他们因有向无环图（DAG）中的安全博弈而研究该问题。我们关注 $\pi$ 具有仿射形式的情况：$\pi_P = 1 - \sum_{e \in P} \mu_e$，其中 $\mu \in [0, 1]^E$。期望分布存在的必要条件是：对所有 $P \in \mathcal{P}$，有 $\sum_{e \in P} \rho_e \geq \pi_P$。我们证明，该条件为充分条件当且仅当 $\mathcal{P}$ 具有弱最大流/最小割性质。我们进一步在 $(E, \mathcal{P})$ 为抽象网络的特殊情况下，提供了计算相应分布的高效组合算法。作为推论，Dahan 等人的安全博弈均衡可在广泛场景（包括任意有向图）中高效计算。作为算法的子程序，我们提供了在抽象网络中计算最短路径的组合算法，部分回答了 McCormick (SODA 1996) 提出的开放问题。我们进一步证明，Dahan 等人针对有向无环图中需求向量 $\pi$ 提出的守恒律可简化为上述仿射需求设定。