We consider a general inference setting for discrete probabilistic graphical models where we seek maximum a posteriori (MAP) estimates for a subset of the random variables (max nodes), marginalizing over the rest (sum nodes). We present a hybrid message-passing algorithm to accomplish this. The hybrid algorithm passes a mix of sum and max messages depending on the type of source node (sum or max). We derive our algorithm by showing that it falls out as the solution of a particular relaxation of a variational framework. We further show that the Expectation Maximization algorithm can be seen as an approximation to our algorithm. Experimental results on synthetic and real-world datasets, against several baselines, demonstrate the efficacy of our proposed algorithm.