We give an optimal solution to the Maximum All Request Path Grooming (MARPG) problem motivated by a traffic grooming application. 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). We present a greedy algorithm to solve the problem and an explicit formula for the maximum number of requests that can be groomed. In particular, if C = s(s+1)/2 and N \ge s(s-1), an optimal solution is obtained by taking all the requests of smallest length, that is of length 1 to s. However this is not true in general since anomalies can exist. We give a complete analysis and the exact number of such anomalies.