  We describe a new dependent-rounding algorithmic framework for bipartite
graphs. Given a fractional assignment $\vec x$ of values to edges of graph $G =
(U \cup V, E)$, the algorithms return an integral solution $\vec X$ such that
each right-node $v \in V$ has at most one neighboring edge $f$ with $X_f = 1$,
and where the variables $X_e$ also satisfy broad nonpositive-correlation
properties. In particular, for any edges $e_1, e_2$ sharing a left-node $u \in
U$, the variables $X_{e_1}, X_{e_2}$ have strong negative-correlation
properties, i.e. the expectation of $X_{e_1} X_{e_2}$ is significantly below
$x_{e_1} x_{e_2}$.
  This algorithm is based on generating negatively-correlated Exponential
random variables and using them in a contention-resolution scheme inspired by
an algorithm Im & Shadloo (2020). Our algorithm gives stronger and much more
flexible negative correlation properties.
  Dependent rounding schemes with negative correlation properties have been
used for approximation algorithms for job-scheduling on unrelated machines to
minimize weighted completion times (Bansal, Srinivasan, & Svensson (2021), Im &
Shadloo (2020), Im & Li (2023)). Using our new dependent-rounding algorithm,
among other improvements, we obtain a $1.398$-approximation for this problem.
This significantly improves over the prior $1.45$-approximation ratio of Im &
Li (2023).
