Optimal Union Probability Interval Is NP-Hard

2026-05-05

AI Summary

This study addresses the classical problem in probabilistic reasoning of determining the tightest possible bounds on the probability of the union of events when only partial intersection probabilities are known. Adopting a geometric perspective, the authors reformulate the problem as a linear programming projection over the atoms of a Venn diagram and uncover deep connections to cut polytopes, correlation polytopes, and marginal polytopes arising in graphical models. For the first time, they rigorously establish that this problem is NP-hard, thereby resolving a long-standing theoretical gap noted by Pitowsky and Boros et al. This result provides a firm computational complexity lower bound, offering foundational hardness guarantees for related inference tasks in probabilistic logic and machine learning.

Abstract

A problem dating back to Boole [Laws of Thought, Walton & Maberly,1854] is what can be computed about the probability of a finite union of events when given as input the probabilities of intersections of some of the events. The modern geometric study of the problem can be traced back to Hailperin [Amer. Math. Monthly 2 (1965) 343--359] who phrased the problem in the language of linear programming and generalized it to logical formulas of the events other than disjunction, heralding a substantial body of work in probabilistic logic [Nilsson, Artif.\ Intell.\ 28 (1986) 71--87], including the probabilistic satisfiability problem of Georgakopoulos, Kavvadis, and Papadimitriou [J.Complexity 4 (1988) 1--11], as well as fundamental connections to the geometry of metrics via cut and correlation polytopes [Deza and Laurent, Geometry of Cuts and Metrics, Springer, 1997] and to the study of marginal polytopes in graphical models of machine learning [Wainwright and Jordan, Found.\ Trends Mach.\ Learn. 1 (2008) 1--305]. This paper (i) describes the pertinent geometry of Boole's problem via coordinate projections of an elementary polytope arising essentially from Hailperin's linear program on the atoms of a Venn diagram, and (ii) shows that computing the optimal interval for the union probability is NP-hard, resolving an apparent gap in the literature highlighted by Pitowsky [Math.\ Programming 50 (1991) 395--414] and Boros et al. [Math.\ Oper.\ Res. 39 (2014) 1311--1329 and 51 (2026) 134--148].

Problem

• union probability
• probabilistic logic
• NP-hard
• Boole's problem
• marginal polytope

Innovation

• union probability
• NP-hard
• probabilistic logic
• polytope projection
• Boole's problem