Optimizing pred(25) Is NP-Hard

Usually, in data processing, to find the parameters of the models that best fits the data, people use the Least Squares method. One of the advantages of this method is that for linear models, it leads to an easy-to-solve system of linear equations. A limitation of this method is that even a single outlier can ruin the corresponding estimates; thus, more robust methods are needed. In particular, in software engineering, often, a more robust pred(25) method is used, in which we maximize the number of cases in which the model’s prediction is within the 25% range of the observations. In this paper, we show that even for linear models, pred(25) parameter estimation is NP-hard. 1 Formulation of the Problem Need to estimate parameters of models. In many practical situations, we know that a quantity y depends on the quantities x1, . . . , xn, and we know the general type of this dependence. In precise terms, this means that we know a family of functions f(c1, . . . , cp, x1, . . . , xn) characterized by parameters ci, and we know that the actual dependence corresponds to one of these functions. For example, we may know that the dependence is linear; in this cases, the corresponding family takes the form f(c1, . . . , cn, cn+1, x1, . . . , xn) = cn+1 + n ∑