Computing Standard-Deviation-to-Mean and Variance-to-Mean Ratios under Interval Uncertainty is NP-Hard

Once we have a collection of values corresponding a class of objects, a usual way to decide whether a new object with the value of the corresponding property belongs to this class is to check whether this value belongs to interval from mean E minus k sigma σ to mean plus k sigma, where the parameter k is determined by the degree of confidence with which we want to make the decision. For each value x, the degree of confidence that x belongs to the class depends on the smallest value k for which x belongs to the corresponding interval, i.e., on the ratio r of σ and |E − x|. In practice, we often only know the intervals that contain the actual values. Different values from these intervals lead, in general, to different values of r, so it is desirable to compute the range of corresponding values of r. Polynomial-time algorithms are known for computing this range under certain conditions; whether it is always possible to compute this range in polynomial time was unknown. In this paper, we prove that the problem of computing this range is NP-hard. A similar NP-hardness result is proven for a similar ratio between the variance V and the mean E which is used in clustering. c ©2015 World Academic Press, UK. All rights reserved.