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

TL;DR

This paper characterizes when a probability distribution satisfying certain marginals and security constraints exists in max-flow/min-cut systems, providing an efficient algorithm for a special case and connecting to security game equilibria.

Contribution

It establishes a necessary and sufficient condition for the existence of such distributions based on the weak max-flow/min-cut property and offers an efficient combinatorial algorithm for abstract networks.

Findings

Condition for distribution existence is both necessary and sufficient.

Efficient algorithm for abstract network cases.

Connection to security game equilibria and shortest path algorithms.

Abstract

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.