In Answer Set Programming (ASP), the user can define declaratively a problem and solve it with efficient solvers; practical applications of ASP are countless and several constraint problems have been successfully solved with ASP. On the other hand, solution time usually grows in a superlinear way (often, exponential) with respect to the size of the instance, which is impractical for large instances. A widely used approach is to split the optimization problem into sub-problems that are solved in sequence, some committing to the values assigned by others, and reconstructing a valid assignment for the whole problem by juxtaposing the solutions of the single sub-problems. On the one hand this approach is much faster, due to the superlinear behavior; on the other hand, it does not provide any guarantee of optimality: committing to the assignment of one sub-problem can rule out the optimal solution from the search space. In other research areas, Logic-Based Benders Decomposition (LBBD) proved effective; in LBBD, the problem is decomposed into a Master Problem (MP) and one or several Sub-Problems (SP). The solution of the MP is passed to the SPs, that can possibly fail. In case of failure, a no-good is returned to the MP, that is solved again with the addition of the new constraint. The solution process is iterated until a valid solution is obtained for all the sub-problems or the MP is proven infeasible. The obtained solution is provably optimal under very mild conditions. In this paper, we apply for the first time LBBD to ASP, exploiting an application in health care as case study. Experimental results show the effectiveness of the approach. We believe that the availability of LBBD can further increase the practical applicability of ASP technologies.