The fundamental problem for inference control in data cubes is how to efficiently calculate the lower and upper bounds for each cell value given the aggregations of cell values over multiple dimensions. In this paper, we provide the first practical solution for estimating exact bounds in two-dimensional irregular data cubes (i.e., data cubes in which certain cell values are known to a snooper). Our results imply that the exact bounds cannot be obtained by a direct application of the Fr?echet bounds in some cases. We then propose a new approach to improve the classic Fr?echet bounds for any high-dimensional data cube in the most general case. The proposed approach improves upon the Fr?echet bounds in the sense that it gives bounds that are at least as tight as those computed by Fr?echet, yet is simpler in terms of time complexity. Based on our solutions to the fundamental problem, we discuss two security applications, privacy protection of released data and fine-grained access control and auditing.