We propose a new approach based on model relaxation to compute minimum-cardinality diagnoses of a (faulty) system: We obtain a relaxed model of the system by splitting nodes in the system and compile the abstraction of the relaxed model into DNNF. Abstraction is obtained by treating self-contained sub-systems called cones as single components. We then use a novel branch-and-bound search algorithm and compute the abstract minimum-cardinality diagnoses of the system, which are later refined hierarchically, in a careful manner, to get all minimum-cardinality diagnoses of the system. Experiments on ISCAS-85 benchmark circuits show that the new approach is faster than the previous state-of-the-art hierarchical approach, and scales to all circuits in the suite for the first time.