1 Finishing the proof of Rent – or – Buy Problem 6

Last time we talked the Rent-or-Buy problem, where we are given a graph and a set of demands from the source to terminal nodes. We want to satisfy the demands by connecting terminals to the source (think about sending flows), and we need to either “buy” or “rent” edges used in these paths. Every edge has an associated cost ce: when renting the edge, we need to pay ce more for using the edge on paths connecting each new source–terminal pair; when buying the edge, we pay M · ce for using the edge on whichever number of source–terminal pairs. The constant M is the same for all edges. If we have decided on paths for connecting all source–terminal pairs, then the number of paths using each edge is fixed, and we would rent or buy by choosing the lower price. However, there could be many ways to wire the paths between source-terminal pairs, so we want to find out the cheapest wiring and buy or rent accordingly.