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- W4308759323 abstract "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." @default.
- W4308759323 created "2022-11-15" @default.
- W4308759323 creator A5055549371 @default.
- W4308759323 date "2022-11-09" @default.
- W4308759323 modified "2023-09-27" @default.
- W4308759323 title "Decomposition of Probability Marginals for Security Games in Abstract Networks" @default.
- W4308759323 doi "https://doi.org/10.48550/arxiv.2211.04922" @default.
- W4308759323 hasPublicationYear "2022" @default.
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