Short length versions of Menger's theorem

Consider a simple n-vertex undirected graph and assume there are ~ edge-disjoint paths between two vertices u and V. We prove the following two results: There are K edge-disjoint paths between u and v, the average length of which is 0(7t/~) If all vertices have degree at least HI there are ~ edge-disjoint paths between u and v, each of which has length O(n/K). These bouncls are best possible. For directed graphs, the first result still holds but not the second. Some of the paths can be at least Q(n) long. We also describe how to use a minimum cost flow algorithm to find the paths irnpliecl by the above results in time O(tcrn). In a ~ edge-connected graph, we define the concept of B-distance (or bulk distance). The B-distance between u and v is the minimum over all R edge-disjoint paths between u and v of the maximum path length. We prove that B-distance forms a metric. We give NPhardness results on computing B-distances in two cases. The third remaining case is open but we give evidence to its difficulty.