An algorithmic Friedman--Pippenger theorem on tree embeddings and applications to routing
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
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Given a digraph D=(V,A) and a set of $\kappa$ pairs of vertices in V, we are interested in finding, for each pair (xi, yi), a directed path connecting xi to yi such that the set of $\kappa$ paths so found is arc-disjoint. For arbitrary graphs the problem is ${\cal NP}$-complete, even for $\kappa=2$. 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 the degree r is sufficiently large, then all sets of $\kappa=\Omega(n/\log n)$ pairs of vertices can be joined. This is within a constant factor of best possible.