MINIMUM SPANNING TREES IN INFINITE GRAPHS: THEORY AND ALGORITHMS\ast

We discuss finding minimum spanning trees (MSTs) on connected graphs with countably many nodes of finite degree. When edge costs are summable and an MST exists (which is not guaranteed in general), we show that an algorithm that finds MSTs on finite subgraphs (called layers) converges in objective value to the cost of an MST of the whole graph as the sizes of the layers grow to infinity. We call this the layered greedy algorithm since a greedy algorithm is used to find MSTs on each finite layer. We stress that the overall algorithm is not greedy since edges can enter and leave iterate spanning trees as larger layers are considered. However, in the setting where the underlying graph has the finite cycle (FC) property (meaning that every edge is contained in at most finitely many cycles) and distinct edge costs, we show that a unique MST T\ast exists and the layered greedy algorithm produces iterates that converge to T\ast by eventually ``locking in"" edges after finitely many iterations. Applications to network deployment are discussed.