Many real-world networks face a fundamental tension: should you invest upfront in shared infrastructure, or pay as you go? An oil company routing pipelines from remote wells to a refinery faces exactly this choice: install high-capacity pipes (expensive but reusable) or lease smaller capacity (cheap per unit, but costs accumulate with usage). The same structure appears in telecommunications infrastructure, VLSI design, and fiber-optic network planning. Connected Facility Location (CFL) problem formalizes this tension. CFL is NP-hard, so the question is: how close to optimal can a polynomial-time algorithm get? We studied this question by analyzing a clever but natural primal dual formulation for the Single-Sink Rent-or-Buy (SRoB)special case of CFL. The LP relaxation produces a polynomial time moat-growing algorithm that produces optimal solution for many cases. Our central contribution was two classes of pathological examples where this algorithm fails arbitrarily badly. In both cases, a large group of clients merges early and collectively “buys” edges that are only useful to other clients: overpaying on their behalf. A bidirected cut relaxation fixes the first class, but a subtler general graph example defeats even that.