The Complexity of Finding kth Most Probable Explanations in Probabilistic Networks

  • Johan H. P. Kwisthout
  • Hans L. Bodlaender
  • Linda C. van der Gaag
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6543)


In modern decision-support systems, probabilistic networks model uncertainty by a directed acyclic graph quantified by probabilities. Two closely related problems on these networks are the Kth MPE and Kth Partial MAP problems, which both take a network and a positive integer k for their input. In the Kth MPE problem, given a partition of the network’s nodes into evidence and explanation nodes and given specific values for the evidence nodes, we ask for the kth most probable combination of values for the explanation nodes. In the Kth Partial MAP problem in addition a number of unobservable intermediate nodes are distinguished; we again ask for the kth most probable explanation. In this paper, we establish the complexity of these problems and show that they are FP PP - and \({\mathsf{FP^{\mathsf{PP^{\mathsf{PP}}}}}}\)-complete, respectively.


Intermediate Node Turing Machine Probable Explanation Classical Swine Fever Virus Explanation Node 
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  1. 1.
    Charitos, T., van der Gaag, L.C., Visscher, S., Schurink, C.A.M., Lucas, P.J.F.: A dynamic Bayesian network for diagnosing ventilator-associated pneumonia in ICU patients. Expert Systems with Applications 26, 1249–1258 (2009)CrossRefGoogle Scholar
  2. 2.
    Geenen, P.L., Elbers, A.R.W., van der Gaag, L.C., Loeffen, W.L.A.: Development of a probabilistic network for clinical detection of classical swine fever. In: Proceedings of the 11th Symposium of the International Society for Veterinary Epidemiology and Economics, Cairns, Australia, pp. 667–669 (2006)Google Scholar
  3. 3.
    Pearl, J.: Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, Palo Alto (1988)zbMATHGoogle Scholar
  4. 4.
    Roth, D.: On the hardness of approximate reasoning. Artificial Intelligence 82, 273–302 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Shimony, S.E.: Finding MAPs for belief networks is NP-hard. Artificial Intelligence 68, 399–410 (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L., van den Eijkhof, F., van der Gaag, L.C.: On the complexity of the MPA problem in probabilistic networks. In: van Harmelen, F. (ed.) Proceedings of the 15th European Conference on Artificial Intelligence, pp. 675–679. IOS Press, Amsterdam (2002)Google Scholar
  7. 7.
    Park, J.D., Darwiche, A.: Complexity results and approximation settings for MAP explanations. Journal of Artificial Intelligence Research 21, 101–133 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Charniak, E., Shimony, S.E.: Cost-based abduction and MAP explanation. Acta Informatica 66, 345–374 (1994)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Abdelbar, A., Hedetniemi, S.: Approximating MAPs for belief networks is NP-hard and other theorems. Artificial Intelligence 102, 21–38 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jensen, F.V., Nielsen, T.D.: Bayesian Networks and Decision Graphs. Springer, New York (2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    Papadimitriou, C.H.: On the complexity of unique solutions. Journal of the ACM 31, 392–400 (1984)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Krentel, M.W.: The complexity of optimization problems. Journal of Computer and System Sciences 36, 490–509 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Toda, S.: Simple characterizations of P(#P) and complete problems. Journal of Computer and System Sciences 49, 1–17 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kwisthout, J.H.P.: The Computational Complexity of Probabilistic Networks. PhD thesis, Universiteit Utrecht (2009)Google Scholar
  15. 15.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Johan H. P. Kwisthout
    • 1
  • Hans L. Bodlaender
    • 1
  • Linda C. van der Gaag
    • 1
  1. 1.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands

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