The recent development in network multimedia technology has created numerous real-time multimedia applications where the Quality-of-Service (QoS) requirements are quite rigorous. This has made multicasting under QoS constraints one of the most prominent routing problems. The authors consider the problem of the efficient delivery of data stream to receivers for multi-source communication groups. Efficiency in this context means to minimize cost while meeting bounds on the end-to-end delay of the application. The authors adopt the multi-core approach and utilize SPAN (Karaman and Hassane, 2007)—a core-based framework for multi-source group applications — as the basis to develop greedy randomized adaptive search procedures (GRASP) for the associated constrained cost minimization problem. The procedures are tested in asymmetric networks and computational results show that they consistently outperform their counterparts in the literature.