A Fast 5/2-Approximation Algorithm for Hierarchical Scheduling

We present in this article a new approximation algorithm for scheduling a set of

n

independent rigid (meaning requiring a fixed number of processors) jobs on hierarchical parallel computing platform. A hierarchical parallel platform is a collection of

k

parallel machines of different sizes (number of processors). The jobs are submitted to a central queue and each job must be allocated to one of the

k

parallel machines (and then scheduled on some processors of this machine), targeting the minimization of the maximum completion time (makespan). We assume that no job require more resources than available on the smallest machine.

This problem is hard and it has been previously shown that there is no polynomial approximation algorithm with a ratio lower than 2 unless

P

 = 

NP

. The proposed scheduling algorithm achieves a

${{5}\over{2}}$

ratio and runs in

O

(

log

(

np

max

)

knlog

(

n

)), where

p

max

is the maximum processing time of the jobs. Our results also apply for the Multi Strip Packing problem where the jobs (rectangles) must be allocated on contiguous processors.