Finding MAPs for belief networks is NP-hard

Research output: Contribution to journalArticlepeer-review

240 Scopus citations


Given a probabilistic world model, an important problem is to find the maximum a-posteriori probability (MAP) instantiation of all the random variables given the evidence. Numerous researchers using such models employ some graph representation for the distributions, such as a Bayesian belief network. This representation simplifies the complexity of specifying the distributions from exponential in n, the number of variables in the model, to linear in n, in many interesting cases. We show, however, that finding the MAP is NP-hard in the general case when these representations are used, even if the size of the representation happens to be linear in n. Furthermore, minor modifications to the proof show that the problem remains NP-hard for various restrictions of the topology of the graphs. The same technique can be applied to the results of a related paper (by Cooper), to further restrict belief network topology in the proof that probabilistic inference is NP-hard.

Original languageEnglish
Pages (from-to)399-410
Number of pages12
JournalArtificial Intelligence
Issue number2
StatePublished - 1 Jan 1994


  • Abductive reasoning
  • Complexity
  • Diagnosis
  • Explanation
  • Probabilistic reasoning

ASJC Scopus subject areas

  • Language and Linguistics
  • Linguistics and Language
  • Artificial Intelligence


Dive into the research topics of 'Finding MAPs for belief networks is NP-hard'. Together they form a unique fingerprint.

Cite this