  In this paper, we consider the online problem of scheduling independent jobs
\emph{non-preemptively} so as to minimize the weighted flow-time on a set of
unrelated machines. There has been a considerable amount of work on this
problem in the preemptive setting where several competitive algorithms are
known in the classical competitive model. %Using the speed augmentation model,
Anand et al. showed that the greedy algorithm is
$O\left(\frac{1}{\epsilon}\right)$-competitive in the preemptive setting. In
the non-preemptive setting, Lucarelli et al. showed that there exists a strong
lower bound for minimizing weighted flow-time even on a single machine.
However, the problem in the non-preemptive setting admits a strong lower bound.
Recently, Lucarelli et al. presented an algorithm that achieves a
$O\left(\frac{1}{\epsilon^2}\right)$-competitive ratio when the algorithm is
allowed to reject $\epsilon$-fraction of total weight of jobs and
$\epsilon$-speed augmentation. They further showed that speed augmentation
alone is insufficient to derive any competitive algorithm. An intriguing open
question is whether there exists a scalable competitive algorithm that rejects
a small fraction of total weights.
  In this paper, we affirmatively answer this question. Specifically, we show
that there exists a $O\left(\frac{1}{\epsilon^3}\right)$-competitive algorithm
for minimizing weighted flow-time on a set of unrelated machine that rejects at
most $O(\epsilon)$-fraction of total weight of jobs. The design and analysis of
the algorithm is based on the primal-dual technique. Our result asserts that
alternative models beyond speed augmentation should be explored when designing
online schedulers in the non-preemptive setting in an effort to find provably
good algorithms.
