Decomposition of Probability Marginals for Security Games in Abstract Networks

Given a set system $(E, \mathcal{P})$, let $\pi \in [0,1]^{\mathcal{P}}$ be a vector of requirement values on the sets and let $\rho \in [0, 1]^E$ be a vector of probability marginals with $\sum_{e \in P} \rho_e \geq \pi_P$ for all $P \in \mathcal{P}$. We study the question under which conditions the marginals $\rho$ can be decomposed into a probability distribution on the subsets of $E$ such that the resulting random set intersects each $P \in \mathcal{P}$ with probability at least $\pi_P$. Extending a result by Dahan, Amin, and Jaillet (MOR 2022) motivated by a network security game in directed acyclic graphs, we show that such a distribution exists if $\mathcal{P}$ is an abstract network and the requirements are of the form $\pi_P = 1 - \sum_{e \in P} \mu_e$ for some $\mu \in [0, 1]^E$. Our proof yields an explicit description of a feasible distribution that can be computed efficiently. As a consequence, equilibria for the security game studied by Dahan et al. can be efficiently computed even when the underlying digraph contains cycles. As a subroutine of our algorithm, we provide a combinatorial algorithm for computing shortest paths in abstract networks, answering an open question by McCormick (SODA 1996). We further show that a conservation law proposed by Dahan et al. for requirement functions in partially ordered sets can be reduced to the setting of affine requirements described above.