The fundamental Quality of Service (QoS) routing problem, which consists in determining paths subject to multiple QoS constraints, has been extensively investigated in the intra-domain context. However, few solutions exist for the inter-domain case, despite the importance of this problem to enable the delivery of services with QoS across domain boundaries. We propose a method that distributes the operations to compute inter-domain constrained paths. This method relies on a per-domain formulation that is compatible with the path computation element framework. It enables us to propose the first algorithm that guarantees to find optimal paths subject to an arbitrary number of constraints. These paths ensure the best QoS performance with respect to the constraints.