We present an optimization problem that arises in the context of system-design and the refinement of high-level specifications. We present a graph-based formalization of the problem, thus defining a new optimization problem, which we call MINIMUM CONFIGURATION. We prove MINIMUM CONFIGURATION to be NP-hard and also hard to approximate, i.e. not to be in APX. These results are obtained by reduction from graph coloring and SHORTEST PATH WITH FORBIDDEN PAIRS. We show how to apply the introduced

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... epts to the application domain and propose a way to identify subclasses in the input space, for which MINIMUM CONFIGURATION can be solved efficiently.