  We discuss one of the most fundamental scheduling problem of processing jobs
on a single machine to minimize the weighted flow time (weighted response
time). Our main result is a $O(\log P)$-competitive algorithm, where $P$ is the
maximum-to-minimum processing time ratio, improving upon the
$O(\log^{2}P)$-competitive algorithm of Chekuri, Khanna and Zhu (STOC 2001). We
also design a $O(\log D)$-competitive algorithm, where $D$ is the
maximum-to-minimum density ratio of jobs. Finally, we show how to combine these
results with the result of Bansal and Dhamdhere (SODA 2003) to achieve a
$O(\log(\min(P,D,W)))$-competitive algorithm (where $W$ is the
maximum-to-minimum weight ratio), without knowing $P,D,W$ in advance. As shown
by Bansal and Chan (SODA 2009), no constant-competitive algorithm is achievable
for this problem.
