Minimizing General Delay Costs on Unrelated Machines

We consider the problem of minimizing general delay costs on unrelated machines. In this problem, we are given a set of m machines and a set of n jobs. Each job j has a machine dependent processing time of pij on machine i, and a release time rij . Further, we are also given a non-decreasing function fj(t) which gives the cost of completing the job j at time t. Our goal is to assign each job to exactly one machine and specify a scheduling policy on each machine such that ∑ j fj(Cj) is minimized, where Cj is the completion time of job j. This objective function captures several widely studied objective functions such as makespan, average completion time/flow-time, tardiness, and deadline scheduling. We design the first polynomial time bicriteria approximation algorithm for the problem that achieves a constant approximation when given a small constant factor speed augmentation. The speed augmentation is necessary as the objective function value can be zero, and hence no algorithm can achieve a bounded approximation factor without speed augmentation. As a corollary of our result, we also get the first polynomial time approximation algorithm for the tardiness objective for unrelated machines (or even for identical machines). ∗Microsoft Research, Redmond, [email protected] †Microsoft Research, Redmond, [email protected]