We give time lower bounds for the distributed approximation of minimum vertex cover (MVC) and related problems such as minimum dominating set (MDS). In k communication rounds, MVC and MDS can only be approximated by factors Ω(n c/k 2 /k) and Ω(∆ 1/k /k) for some constant c, where n and ∆ denote the number of nodes and the largest degree in the graph. The number of rounds required in order to achieve a constant or even only a polylogarithmic approximation ratio is at least Ω( Ô log n/ log log n)

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... and Ω(log ∆/ log log ∆). By a simple reduction, the latter lower bounds also hold for the construction of maximal matchings and maximal independent sets.