This paper makes the case for a parametrized complexity approach to tackle the fundamental but notoriously hard Virtual Network Embedding Problem. In particular, we show that the structure of the to-be-embedded virtual network requests can be exploited toward fast (i.e.,fixed-parameter tractable) approximation algorithms, using dynamic as well as linear programming algorithms. Our approach does provide formal guarantees on the runtime and solution quality and can safeguard also latency constraints. Using extensive computational experiments we demonstrate the practical relevance of our novel approach.