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(logP)-competitive algorithm, where P is the
maximum-to-minimum processing time ratio, improving upon the
O(log2P)-competitive algorithm of Chekuri, Khanna and Zhu (STOC 2001). We
also design a O(logD)-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.Comment: 20 pages, 4 figure