We give an optimal solution to the Maximum All Request Path Grooming (MARPG) problem motivated by a traffic grooming application and for its interest for computing lower bounds for the cutwidth of a graph. The MARPG problem consists in finding the maximum number of connections which can be established in a path of size n, where each arc has a capacity or bandwidth C (grooming factor). A simple greedy solution is obtained by taking all requests of smallest length. However this is not optimal in general since anomalies can exist. We give a complete analysis and the exact number of such anomalies. Then, we establish a formula returning in constant time the value of the optimal solution to the MARPG problem. We also propose a polynomial time algorithm to get the set of requests of an optimal solution. Finally, we establish upper bounds on the number of anomalies, for example the general one is (5 -2 √ 6)C. These bounds can be used to get new lower bounds for the cutwidth of a graph.