We consider the problem of minimizing average flow time on multiple machines when each job can be assigned only to a specified subset of the machines. This is a special case of scheduling on unrelated machines and we show that no online algorithm can have a bounded competitive ratio. We provide an {\rm O}(\log P)-approximation algorithm by modifying the single-source unsplittable flow algorithm of Dinitz et.al. Here P is the ratio of the maximum to the minimum processing times.

We establish an \Omega (\log P)-integrality gap for our LPrelaxation and use this to show an \Omega (\log P/\log \log P) lower bound on the approximability of the problem. We then extend the hardness results to the problem of minimizing flow time on parallel machines and establish the first non-trivial lower bounds on the approximability; we show that the problem cannot be approximated to within \Omega (\sqrt {\log P/\log \log P} ).