Reliable Computing

, Volume 13, Issue 3, pp 261–282

# Unimodality, Independence Lead to NP-Hardness of Interval Probability Problems

• Daniel J. Berleant
• Olga Kosheleva
• Hung T. Nguyen
Article

## Abstract

In many real-life situations, we only have partial information about probabilities. This information is usually described by bounds on moments, on probabilities of certain events, etc. –i.e., by characteristics c(p) which are linear in terms of the unknown probabilities p j. If we know interval bounds on some such characteristics $$\underline{a}_i\leq c_i(p)\leq \bar{a}_i$$, and we are interested in a characteristic c(p), then we can find the bounds on c(p) by solving a linear programming problem.

In some situations, we also have additional conditions on the probability distribution –e.g., we may know that the two variables x 1 and x 2 are independent, or that the joint distribution of x 1 and x 2 is unimodal. We show that adding each of these conditions makes the corresponding interval probability problem NP-hard.

## Keywords

Linear Programming Problem Linear Constraint Stochastic Dominance Unimodal Distribution Integer Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Daniel J. Berleant
• 1
• Olga Kosheleva
• 2