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George Boole first proposed the union bounding problem, which is a class ofprobabilistic satisfiability problems. Based on Hailperin's linear programmingformulation of the problem, a very large body of research was conducted toprovide good bounds by considering a relaxation that is either polynomialdimension in $n$ or is at least polynomially computable. We study monotonelinear mappings $\mathcal{L}$ to a lower $N$ dimensional space such that$\mathcal{L}(\mathbb{R}_+^{2^n-1}) \subseteq \mathbb{R}_+^N$ and provide acomplete characterization of the cone $L_\mathcal{E}(\mathbb{R}_+^\Omega)$ forthree aggregation models from the literature. This characterization leads tostrictly improved bounds in two of these cases as a result. We also obtain an$O(n^2)$ dimensional linear program that is equivalent with Hailperin's model.An interesting side result is that there exists a connection betweenHailperin's model and the cut-polytope.

Boole's probability bounding problem, linear programming aggregations, and nonnegative quadratic pseudo-Boolean functions