  We give deterministic distributed $(1+\epsilon)$-approximation algorithms for
Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the
LOCAL model. Our coloring algorithm runs in $O(\frac{1}{\epsilon} \log n)$
rounds, and our independent set algorithm has a runtime of
$O(\frac{1}{\epsilon}\log(\frac{1}{\epsilon})\log^* n)$ rounds. For coloring,
existing lower bounds imply that the dependencies on $\frac{1}{\epsilon}$ and
$\log n$ are best possible. For independent set, we prove that
$O(\frac{1}{\epsilon})$ rounds are necessary.
  Both our algorithms make use of a tree decomposition of the input chordal
graph. They iteratively peel off interval subgraphs, which are identified via
the tree decomposition of the input graph, thereby partitioning the vertex set
into $O(\log n)$ layers. For coloring, each interval graph is colored
independently, which results in various coloring conflicts between the layers.
These conflicts are then resolved in a separate phase, using the particular
structure of our partitioning. For independent set, only the first $O( \log
\frac{1}{\epsilon})$ layers are required as they already contain a large enough
independent set. We develop a $(1+\epsilon)$-approximation maximum independent
set algorithm for interval graphs, which we then apply to those layers.
  This work raises the question as to how useful tree decompositions are for
distributed computing.
