Probabilistic reasoning in intelligent systems: networks of plausible inference
Probabilistic reasoning in intelligent systems: networks of plausible inference
Complexity classes defined by counting quantifiers
Journal of the ACM (JACM)
Finding MAPs for belief networks is NP-hard
Artificial Intelligence
On the hardness of approximate reasoning
Artificial Intelligence
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Stochastic Boolean Satisfiability
Journal of Automated Reasoning
Monotonicity in Bayesian networks
UAI '04 Proceedings of the 20th conference on Uncertainty in artificial intelligence
Complexity results and approximation strategies for MAP explanations
Journal of Artificial Intelligence Research
Probabilities for a probabilistic network: a case study in oesophageal cancer
Artificial Intelligence in Medicine
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Many computational problems related to probabilistic networks are complete for complexity classes that have few 'real world' complete problems. For example, the decision variant of the inference problem (pr) is PP-complete, the map-problem is NPPP-complete and deciding whether a network is monotone in mode or distribution is CONPPP- complete. We take a closer look at monotonicity; more specific, the computational complexity of determining whether the values of the variables in a probabilistic network can be ordered, such that the network is monotone. We prove that this problem - which is trivially co-NPPP-hard - is complete for the class co-NPNPPP in networks which allow implicit representation.