In the classical vertex cover problem, we are given a graph G=(V,E) and we aim to find a minimum cardinality cover of the edges, i.e. a subset of the vertices C⊆V such that for every edge e∈E, at least one of its extremities belongs to C. In the Min-Power-Cover version of the vertex cover problem, we consider an edge-weighted graph and we aim to find a cover of the edges and a valuation (power) of the vertices of the cover minimizing the total power of the vertices. We say that an edge e is covered if at least one of its extremities has a valuation (power) greater than or equal than the weight of e. In this paper, we consider Min-Power-Cover variants of various classical problems, including vertex cover, min cut, spanning tree and path problems.