  We develop deterministic approximation algorithms for the minimum dominating
set problem in the CONGEST model with an almost optimal approximation
guarantee. For $\epsilon>1/{\text{{poly}}}\log \Delta$ we obtain two algorithms
with approximation factor $(1+\epsilon)(1+\ln (\Delta+1))$ and with runtimes
$2^{O(\sqrt{\log n \log\log n})}$ and $O(\Delta\cdot\text{poly}\log \Delta
+\text{poly}\log \Delta \log^{*} n)$, respectively. Further we show how
dominating set approximations can be deterministically transformed into a
connected dominating set in the \CONGEST model while only increasing the
approximation guarantee by a constant factor. This results in a deterministic
$O(\log \Delta)$-approximation algorithm for the minimum connected dominating
set with time complexity
  $2^{O(\sqrt{\log n \log\log n})}$.
