We study two important implications of the constraint composite graph (CCG) associated with the weighted constraint satisfaction problem (WCSP). First, we show that the Nemhauser-Trotter (NT) reduction popularly used for kernelization of the minimum weighted vertex cover (MWVC) problem can also be applied to the CCG of the WCSP. This leads to a polynomial-time preprocessing algorithm that fixes the optimal values of a large subset of the variables in the WCSP. Second, belief propagation (BP) is a well-known technique used for solving many combinatorial problems in probabilistic reasoning, artificial intelligence and information theory. The min-sum message passing (MSMP) algorithm is a simple variant of BP that has also been successfully employed in several research communities. Unfortunately, the MSMP algorithm has met with little success on the WCSP. We revive the MSMP algorithm for solving the WCSP by applying it on the CCG of a given WCSP instance instead of its original form. We refer to this new MSMP algorithm as the lifted MSMP algorithm for the WCSP. We demonstrate the effectiveness of our algorithms through experimental evaluations.