Arc-disjoint paths in expander digraphs

Given a digraph D=(V, A) and a set of κ pairs of vertices in V, we are interested in finding for each pair (x_i, y_i), a directed path connecting x_i to y_i, such that the set of κ paths so found is arc-disjoint. For arbitrary graphs, the problem is 𝒩𝒫-complete, even for κ=2. We present a polynomial time randomized algorithm for finding arc-disjoint paths in an r-regular expander digraph D. We show that if D has sufficiently strong expansion properties and r is sufficiently large, then all sets of κ=Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of best possible.