Multi-Commodity Connected Facility Location

In this paper we define the multi-commodity connected facility location (MCFL) problem, a natural generalization of the well-known multi-commodity rent-or-buy (MROB) and connected facility location (CFL) network design problems. Like in MROB, we wish to send one unit of flow from a set of sources to a set of sinks. This flow is supported by renting or buying edges. For each bought edge e, we pay M ≥ 1 times the cost of e, and we are then free to route an unbounded amount of flow along e. For each rented edge e, we pay its cost times the amount of flow routed along e. However, differently from MROB and similarly to CFL, the rented paths cannot reach and leave the connected components of bought edges at any node. This must happen only at specific nodes, the open facilities, for which we have to pay an opening cost. MCFL models, e.g., a transportation system consisting of several disconnected networks (the bought edges). Users can switch from one network to the other on foot or by bike (rented edges), entering and leaving each network at stations (facilities). We present the first constant approximation algorithm for MCFL. Our algorithm is based on two main ingredients. We use an approximation algorithm for the price-collecting facility location problem to identify pairs that can be cheaply connected with rented edges only. The remaining pairs and the facilities serving them are used to define an MROB instance, on which we run a proper approximation algorithm.