Multicommodity Flow and Circuit Switching

Given a set of request pairs in a network, the problem of routing virtual circuits with low congestion is to connect each pair by a path so that few paths use the same link in the network. We build on an earlier multicommodity ow based approach of Leighton & Rao to show that \short" ow-paths lead to path-selections with \low" congestion. This shows that such good path-selections exist for constant-degree expanders with \strong" expansion, generalizing a result of Broder, Frieze & Upfal. We also show, for in nitely many n, n-vertex undirected graphs Gn along with a set T of connection requests, such that: (a) T is fractionally realizable using ow-paths that impose a (fractional) congestion of at most 1; but (b) any rounding of such a ow to the given set of ow-paths, leads to a congestion of (logn= log logn). This is progress on a long-standing open problem.