We consider a special case of the set cover problem which we call the Preselected Path Multi-Cover problem. Given a graph G with demands on the vertices as well as a family of paths on G with capacities and cost, our goal is to find a minimum-cost multisubset of these paths that cover the demands of all vertices. This problem has applications for example in software testing, where graphs of bounded cyclomatic number appear in a natural way in order to bound the software module complexity. We give a 3-approximation for this problem restricted to graphs with minimum-degree at least 2 and bounded cyclomatic number, and a 5-approximation for graphs with bounded cyclomatic number (with no restriction on the minimum degree).