Boole's probability bounding problem, linear programming aggregations, and nonnegative quadratic pseudo-Boolean functions
Endre Boros, Joonhee Lee
TL;DR
The paper tackles Boole's union-bounding problem by formulating a hierarchy of linear aggregations of Hailperin's LP and proving complete polyhedral descriptions of the induced subcones. It shows that the level-2 aggregation $L_{\mathcal{E}^2}$ is faithful and corresponds to the cone of nonnegative quadratic pseudo-Boolean functions, linking to the cut polytope and enabling a polynomial-time, tightened bound framework via the $QPB^-$ model. The resulting bounds are significantly tighter than previous efficiently computable bounds, as demonstrated by substantial improvements in practice. The work advances both the theory of polyhedral descriptions for probability bounding and the practical computation of sharp bounds for union probabilities, with potential extensions to weighted aggregations and further tightening strategies.
Abstract
Hailperin (1965) introduced a linear programming formulation to a difficult family of problems, originally proposed by Boole (1854,1868). Hailperin's model is computationally still difficult and involves an exponential number of variables (in terms of a typical input size for Boole's problem). Numerous papers provided efficiently computable bounds for the minimum and maximum values of Hailperin's model by using aggregation that is a monotone linear mapping to a lower dimensional space. In many cases the image of the positive orthant is a subcone of the positive orthant in the lower dimensional space, and thus including some of the defining inequalities of this subcone can tighten up such an aggregation model, and lead to better bounds. Improving on some recent results, we propose a hierarchy of aggregations for Hailperin's model and a generic approach for the analysis of these aggregations. We obtain complete polyhedral descriptions of the above mentioned subcones and obtain significant improvements in the quality of the bounds.