Max network flow by expanding at most k edges

Question

Suppose that you manage a road network, modeled by a flow network G=(V, E) with source s (e.g., the residential area), sink t (e.g., the business area), and (positive integer) edge capacities c_e for any edge e in E. A new technology has emerged that can improve the capacity of any roads/edges by 1, and you have the budget to apply it to at most k edges. Design an algorithm to find out if you can improve the overall throughput of the road network (i.e., the max flow value), and if so report the corresponding subset of (up to k) edges. You should also show the correctness and running time of your algorithm.

I can only figure out the case where k=1, but how to find a solution to a general k?