Optimal construction of edge-disjoint paths in random regular graphs

Given a graph G = (V, E) and a set of kappa pairs of vertices in V, we are interested in finding, for each pair (a(i), b(i)), a path connecting a(i) to b(i) such that the set of kappa paths so found is edge-disjoint. (For arbitrary graphs the problem is N P-complete, although it is in P if kappa is fixed.) We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph G(n,r), for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every G(n,r) is such that all sets of kappa = Omega(n / log n) pairs of vertices can be joined. This is within a constant factor of the optimum.