This paper deals with the problem of assigning a partitioned task to a distributed computing system. Considering communication overhead and idle time, we successfully develop a mathematical model to describe the cost function, which is defined to evaluate the task turnaround time, under a genaral model of distributed computing systems. Task assignment is formulated as a DUmapping, which maps a directed acyclic task graph onto an undirected system graph. The search of optimal DU-mapping is NP-complete and is transformed into a state space search problem. An approach called critical sink underestimate is developed to attain an optimal DU-mapping and the most nodes in the state space tree are pruned. Experimental results reveal that this method performs very well due to its close evaluation to the real cost.