  We study approximation algorithms for scheduling problems with the objective
of minimizing total weighted completion time, under identical and related
machine models with job precedence constraints. We give algorithms that improve
upon many previous 15 to 20-year-old state-of-art results. A major theme in
these results is the use of time-indexed linear programming relaxations. These
are natural relaxations for their respective problems, but surprisingly are not
studied in the literature.
  We also consider the scheduling problem of minimizing total weighted
completion time on unrelated machines. The recent breakthrough result of
[Bansal-Srinivasan-Svensson, STOC 2016] gave a $(1.5-c)$-approximation for the
problem, based on some lift-and-project SDP relaxation. Our main result is that
a $(1.5 - c)$-approximation can also be achieved using a natural and
considerably simpler time-indexed LP relaxation for the problem. We hope this
relaxation can provide new insights into the problem.
