In this paper, we study the problem of makespan minimization for the multiprocessorscheduling problem in the presence of communication delays. The communication delaybetween two tasks i and j depends on the distancebetween the two processors on which these two tasks are executed. Lahlou shows that asimple polynomial-time algorithm exists when the length of the schedule is at most two(the problem becomes 𝒩𝒫-complete when the length of the scheduleis at most three). We prove that there is no polynomial-time algorithm with a performanceguarantee of less than 4/3 (unless 𝒫 = 𝒩𝒫) to minimizethe makespan when the network topology is a chain or ring and the precedence graph is abipartite graph of depth one. We also develop two polynomial-time approximation algorithmswith constant ratio dedicated to cases where the processor network admits a limited orunlimited number of processors.
