Uncertainty reduction has recently been introduced in the robust optimization literature as a relevant special case of decisiondependent uncertainty. Herein, we first show that when the uncertainty reduction decisions are constrained, the resulting optimization problem is NP-hard. We further show that relaxing these constraints leads to solving a linear number of deterministic problems in certain special cases and illustrate the numerical relevance of this result. We further provide insights into possible MILP reformulations and the strength of their continuous relaxations.