In this paper, we investigate inverse problems of the interval query problem in application to data mining. Let I be the set of all intervals on U = {1, 2, , n}. Consider an objective function f(I), conditional functions ui(I) on I, and define an optimization problem of finding the interval I maximizing f(I) subject to ui(I) > Ki for given real numbers Ki (i = 1, 2, , h). We propose efficient alogorithms to solve the above optimization problem if the objective function is either additive or quotient, and the conditional functions are additive, where a function f is additive if f(I) = ΣiIf^(i) extending a function f^ on U, and quotient if it is represented as a quotient of two additive functions. We use computational-geometric methods such as convex hull, range searching, and multidimensional divide-and-conquer.

We address the problem of computing various types of expressive tests for decision trees and regression trees. Using expressive tests is promising, because it may improve the prediction accuracy of trees, and it may also provide us some hints on scientific discovery. The drawback is that computing an optimal test could be costly. We present a unified framework to approach this problem, and we revisit the design of efficient algorithms for computing important special cases. We also prove that it is intractable to compute an optimal conjunction or disjunction.