Minimizing Flow-Time on Unrelated Machines

We consider some classical flow-time minimization problems in the unrelated machines setting. In this setting, there is a set of m machines and a set of n jobs, and each job j has a machine dependent processing time of pij on machine i. The flow-time of a job is the amount of time the job spends in a system (its completion time minus its arrival time), and is one of the most natural measure of quality of service. We show the following two results: an $O(min(log2 n, log n log P)) approximation algorithm for minimizing the total flow-time, and an O(log n) approximation for minimizing the maximum flow-time. Here P is the ratio of maximum to minimum job size. These are the first known poly-logarithmic guarantees for both the problems.