In this paper, we study the problem of makespan minimization for the multiprocessor
scheduling problem in the presence of communication delays. The communication delay
between two tasks i and j depends on the distance
between the two processors on which these two tasks are executed. Lahlou shows that a
simple polynomial-time algorithm exists when the length of the schedule is at most two
(the problem becomes 𝒩𝒫-complete when the length of the schedule
is at most three). We prove that there is no polynomial-time algorithm with a performance
guarantee of less than 4/3 (unless 𝒫 = 𝒩𝒫) to minimize
the makespan when the network topology is a chain or ring and the precedence graph is a
bipartite graph of depth one. We also develop two polynomial-time approximation algorithms
with constant ratio dedicated to cases where the processor network admits a limited or
unlimited number of processors.