Decomposing Probability Marginals Beyond Affine Requirements

Consider the triplet $(E,\mathcal{P},\pi )$, where E is a finite ground set, $\mathcal{P}\subseteq 2^E$ is a collection of subsets of E and $\pi :\mathcal{P}\to [0,1]$ is a requirement function. Given a vector of marginals$\rho \in {[0,1]}^E$, our goal is to find a distribution for a random subset $S\subseteq E$ such that $\mathbf{Pr}\left[e\in S\right]={\rho }_e$ for all $e\in E$ and $\mathbf{Pr}\left[P\cap S\ne \mathrm{\varnothing }\right]\ge {\pi }_P$ for all $P\in \mathcal{P}$, or to determine that no such distribution exists.

Generalizing results of Dahan, Amin, and Jaillet [6], we devise a generic decomposition algorithm that solves the above problem when provided with a suitable sequence of admissible support candidates (ASCs). We show how to construct such ASCs for numerous settings, including supermodular requirements, Hoffman-Schwartz-type lattice polyhedra [14], and abstract networks where $\pi$ fulfils a conservation law. The resulting algorithm can be carried out efficiently when $\mathcal{P}$ and $\pi$ can be accessed via appropriate oracles. For any system allowing the construction of ASCs, our results imply a simple polyhedral description of the set of marginal vectors for which the decomposition problem is feasible. Finally, we characterize balanced hypergraphs as the systems $(E,\mathcal{P})$ that allow the perfect decomposition of any marginal vector $\rho \in {[0,1]}^E$, i.e., where we can always find a distribution reaching the highest attainable probability $\mathbf{Pr}\left[P\cap S\ne \mathrm{\varnothing }\right]=min\left\{{\sum }_{e\in P}{\rho }_e,1\right\}$ for all $P\in \mathcal{P}$.