Decomposition of Probability Marginals for Security Games in Max-Flow/Min-Cut Systems

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.