Decomposing Probability Marginals Beyond Affine Requirements

Abstract

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 $\pi : \mathcal {P}\rightarrow [0,1]$ 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 $\textbf{Pr}\left[ e \in S\right] = \rho _e$ for all $e \in E$ and $\textbf{Pr}\left[ P \cap S \ne \emptyset \right] \ge \pi _P$ 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 $\textbf{Pr}\left[ P \cap S \ne \emptyset \right] = \min \left\{ \sum _{e \in P} \rho _e, 1\right\}$ for all $P \in \mathcal {P}$.

Proofs of results marked with $(\clubsuit )$ can be found in the full version [24].