Network utility maximization (NUM) problems provide an important approach to conduct network resource management and to view layering as optimization decomposition. In the existing literature, distributed implementations are typically achieved by the means of the so-called dual decomposition technique. However, the span of decomposition possibilities includes many other elements which thus far have not been fully exploited, such as the use of the primal decomposition technique, the versatile introduction of auxiliary variables, and the potential of multilevel decompositions. This paper presents a systematic framework to exploit the potential of the alternative decomposition structures as a way to obtain different distributed algorithms, each with a different tradeoff among convergence speed, message passing amount and asymmetry, and distributed computation architecture. Many specific applications are considered to illustrate the proposed framework, including resourceconstrained and direct-control rate allocation, and rate allocation among QoS classes and with multipath routing. For each of these applications, the associated generalized NUM formulation is first presented, followed by the development of novel alternative decompositions and numerical experiments on the resulting new distributed algorithms.