We develop deterministic approximation algorithms for the minimum dominating set problem in the CONGEST model with an almost optimal approximation guarantee. For ε > 1/ poly log ∆ we obtain two algorithms with approximation factor (1 + ε)(1 + ln(∆ + 1)) and with runtimes 2 O( √ log n log log n) and O(∆ poly log ∆ + poly log ∆ 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 ∆)-approximation algorithm for the minimum connected dominating set with time complexity 2 O( √ log n log log n) .

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