In this paper, we propose a novel, exact border-based approach that provides an optimal solution for the hiding of sensitive frequent itemsets by (i) minimally extending the original database by a synthetically generated database part - the database extension, (ii) formulating the creation of the database extension as a constraint satisfaction problem, (iii) mapping the constraint satisfaction problem to an equivalent binary integer programming problem, (iv) exploiting underutilized synthetic transactions to proportionally increase the support of non-sensitive itemsets, (v) minimally relaxing the constraint satisfaction problem to provide an approximate solution close to the optimal one when an ideal solution does not exist, and (vi) by using a partitioning in the universe of the items to increase the efficiency of the proposed hiding algorithm. Extending the original database for sensitive itemset hiding is proved to provide optimal solutions to an extended set of hiding problems compared to previous approaches and to provide solutions of higher quality. Moreover, the application of binary integer programming enables the simultaneous hiding of the sensitive itemsets and thus allows for the identification of globally optimal solutions.