We investigate complexity and approximation results on a processor networks where the communication delay depends on the distance between the processors performing tasks. We then prove that there is no heuristic with a performance guarantee smaller than 4/3 for makespan minimization for precedence graph on a large class of processor networks like hypercube, grid, torus, and so forth,with a fixed diameter δ ∈ . We extend complexity results when the precedence graph is a bipartite graph. We also design an efficient polynomial-time O δ2 -approximation algorithm for the makespan minimization on processor networks with diameter δ.