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 flow based approach of Leighton and Rao (1996) to show that short flow 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 et al., 1994). We also show, for infinitely many n, n-vertex undirected graphs G_n along with a set T of connection requests, such that: T is fractionally realizable using flow-paths that impose a (fractional) congestion of at most 1; but any rounding of such a flow to the given set of flow-paths, leads to a congestion of Ω(log n/log log n). This is progress on a long-standing open problem