We examine several decidability questions suggested by questions about all-optical networks, related to the gap between maximal load and number of colors (wavelengths) needed for a legal routing on a fixed graph. We prove the multiple fiber conjecture: for every fixed graph G there is a number L ^G such that in the communication network with L ^G parallel fibers for each edge of G, there is no gap (for any load). We prove that for a fixed graph G the existence of a gap is computable, and give an algorithm to compute it. We develop a decomposition theory for paths, defining the notion of prime sets of paths that are finite building blocks for all loads on a fixed graph. Properties of such decompositions yield our theorems.