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Optimal Union Probability Interval Is NP-Hard

Authors Petteri Kaski , Heikki Mannila , Chandra Kanta Mohapatra

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LIPIcs.ESA.2026.129.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Combinatorial optimization
  • Mathematics of computing → Probability and statistics
  • Theory of computation → Linear programming
Keywords
  • combinatorial probability
  • computational complexity
  • correlation polytope
  • generalized Bonferroni inequalities
  • graphical models
  • inclusion-exclusion
  • intersection probability
  • linear programming
  • union polytope

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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].

Cite As Get BibTex

Petteri Kaski, Heikki Mannila, and Chandra Kanta Mohapatra. Optimal Union Probability Interval Is NP-Hard. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 129:1-129:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.129

Author Details

Petteri Kaski
  • Department of Computer Science, Aalto University, Espoo, Finland
Heikki Mannila
  • Department of Computer Science, Aalto University, Espoo, Finland
Chandra Kanta Mohapatra
  • Department of Computer Science, Aalto University, Espoo, Finland

Funding

  • Mannila, Heikki: Heikki Mannila is supported by the Technology Industries of Finland Centennial Foundation.
  • Mohapatra, Chandra Kanta: Chandra Kanta Mohapatra is supported by Helsinki Institute for Information Technology (HIIT) postdoctoral fellowship.

Acknowledgements

We thank the anonymous reviewers for their insightful comments and careful reading of the manuscript.

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