Arc-disjoint paths in expander digraphs

Given a digraph D=(V, A) and a set of /spl kappa/ pairs of vertices in V, we are interested in finding for each pair (x/sub i/, y/sub i/), a directed path connecting x/sub i/ to y/sub i/, such that the set of /spl kappa/ paths so found is arc-disjoint. For arbitrary graphs, the problem is /spl Nscr//spl Pscr/-complete, even for /spl kappa/=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 /spl kappa/=/spl Omega/(n/log n) pairs of vertices can be joined. This is within a constant factor of best possible.