Scheduling UET-tasks on a star network: complexity and approximation

Abstract

In this article we investigate complexity and approximation on a processor network where the communication delay depends on the distance between the processors performing tasks. We then prove that there is no polynomial-time heuristic with a performance guarantee smaller than 6 5 (respectively 14 13 ) for minimization of the makespan (respectively the total job completion time) on a processor network represented by a star. Moreover, we design an efficient polynomial-time approximation algorithm with a worst-case ratio of four. We also prove that a polynomial-time algorithm exists for a schedule with a length of at most four. Lastly, we generalize all previous results on complexity and approximation.

Scheduling in the presence of processor networks : complexity and approximation

“…Picouleau proved that this problem is N P-complete if the precedence graph is a tree or an out-tree. Recently in [11], the authors proved that there is no polynomial-time algorithm with a performance guarantee smaller than 6 5 for the minimization of the makespan on a processor network represented by a star. This model is close to the master-slave architecture.…”

“…• in the general case, the performance ratio is upper-bounded by m 2 + 3 2 − 1 m , and there exists an instance for which the performance ratio is equal to m 8 + 1 2 [18]; • if the number of processors is even, the upper-bound can be improved to 1 + Moreover, Hwang et al [16] studied approximation list algorithms for scheduling problems where the communication times depend on contention, on a distance function for the tasks involved and on the processors that execute the tasks. The [11], the authors proposed a sophisticated polynomial-time approximation algorithm with a ratio equal to four based on three steps for the problem for the makespan minimization problem on a processor network forming a star.…”