  We study the weighted improper coloring problem, a generalization of
defective coloring. We present some hardness results and in particular we show
that weighted improper coloring is not fixed-parameter tractable when
parameterized by pathwidth. We generalize bounds for defective coloring to
weighted improper coloring and give a bound for weighted improper coloring in
terms of the sum of edge weights. Finally we give fixed-parameter algorithms
for weighted improper coloring both when parameterized by treewidth and maximum
degree and when parameterized by treewidth and precision of edge weights. In
particular, we obtain a linear-time algorithm for weighted improper coloring of
interval graphs of bounded degree.
