Computing best-possible bounds for the distribution of a sum of several variables is NP-hard

摘要

In many real-life situations, we know the probability distribution of two random variables x_1 and x_2, but we have no information about the correlation between x_1 and x_2; what are the possible probability distributions for the sum x_1 + x_2? This question was originally raised by A.N. Kolmogorov. Algorithms exist that provide best-possible bounds for the distribution of x_1 + x_2; these algorithms have been implemented as a part of the efficient software for handling probabilistic uncertainty. A natural question is: what if we have several (n > 2) variables with known distribution, we have no information about their correlation, and we are interested in possible probability distribution for the sum y = x_1 + • • • + x_n? Known formulas for the case n = 2 can be (and have been) extended to this case. However, as we prove in this paper, not only are these formulas not best-possible anymore, but in general, computing the best-possible bounds for arbitrary n is an NP-hard (computationally intractable) problem.