The General Data Protection Regulations (GDPR) entitle individuals to explanations for automated decisions. The form, comprehensibility, and even existence of such explanations remain open problems, investigated as part of explainable AI. We adopt the approach of counterfactual explanations and apply it to decisions made by declarative optimization models. We argue that inverse combinatorial optimization is particularly suited for counterfactual explanations but that the computational difficulties and relatively nascent literature make its application a challenge. To make progress, we address the case of counterfactual explanations that isolate the minimal differences for an individual. We show that under two common optimization functions, full inverse optimization is unnecessary. In particular, we show that for functions of the form of the sum of weighted binary variables, which includes frameworks such as weighted MaxSAT, a solution can be found by solving a slightly modified version of the original optimization model. In contrast, the sum of weighted integer variables can be solved with a binary search over a series of modifications to the original model.