  We consider the classic scheduling problem of minimizing the total weighted
flow-time on a single machine (min-WPFT), when preemption is allowed. In this
problem, we are given a set of $n$ jobs, each job having a release time $r_j$,
a processing time $p_j$, and a weight $w_j$. The flow-time of a job is defined
as the amount of time the job spends in the system before it completes; that
is, $F_j = C_j - r_j$, where $C_j$ is the completion time of job. The objective
is to minimize the total weighted flow-time of jobs.
  This NP-hard problem has been studied quite extensively for decades. In a
recent breakthrough, Batra, Garg, and Kumar presented a {\em pseudo-polynomial}
time algorithm that has an $O(1)$ approximation ratio. The design of a truly
polynomial time algorithm, however, remained an open problem. In this paper, we
show a transformation from pseudo-polynomial time algorithms to polynomial time
algorithms in the context of min-WPFT. Our result combined with the result of
Batra, Garg, and Kumar settles the long standing conjecture that there is a
polynomial time algorithm with $O(1)$-approximation for min-WPFT.
