The Degree-Corrected Stochastic Block Model (DCSBM) is a popular model to generate random graphs with community structure given an expected degree sequence. The standard approach of community detection algorithms based on the DCSBM is to search for the model parameters which are the most likely to have produced the observed network data, via maximum likelihood estimation (MLE). Current techniques for the MLE problem are heuristics and therefore do not guarantee convergence to the optimum. We present mathematical programming formulations and exact solution methods that can provably find the model parameters and community assignments of maximum likelihood given an observed graph. We compare the proposed exact methods with classical heuristic algorithms based on expectation-maximization (EM). The solutions given by exact methods give us a principled way of recognizing when heuristic solutions are sub-optimal and measuring how far they are from optimality.