We consider the replication problem of series-parallel (SP) task graphs where each task may run on more than one processor. The objective of the problem is to minimize the total cost of task execution and interprocessor communication. We call it, the minimum cost replication problem for SP graphs (MCRP-SP). In this paper, we adopt a new communication model where the purpose of replication is to reduce the total cost. The class of applications we consider is computation-intensive applications in which the execution cost of a task is greater than its communication cost. The complexity of MCRP-SP for such applications is proved to be NP-complete. We present a branch-and-bound method to find an optimal solution as well as an approximation approach for suboptimal solution. The numerical results show that such replication may lead to a lower cost than the optimal assignment problem (in which each task is assigned to only one processor) does. The proposed optimal solution has the complexity of O(n22nM), while the approximation solution has O(n4M2), where n is the number of processors in the system and M is the number of tasks in the graph. (Also cross-referenced as UMIACS-TR-93-4.1)