Existence and construction of edge disjoint paths on expander graphs

Given an expander graph G = (V,E) and a set of q disjoint pairs of vertices in V , we are interested in finding for each pair (ai, bi), a path connecting ai to bi, such that the set of q paths so found is edge-disjoint. (For general graphs the related decision problem is NPcomplete.) We prove sufficient conditions for the existence of edge-disjoint paths connecting any set of q ≤ n/(logn)κ disjoint pairs of vertices on any n vertex bounded degree expander, where κ depends only on the expansion properties of the input graph, and not on n. Furthermore, we present a randomized o(n3) time algorithm, and a random NC algorithm for constructing these paths. (Previous existence proofs and construction algorithms allowed only up to nǫ pairs, for some ǫ ≪ 1/3, and strong expanders [19].) In passing, we develop an algorithm for splitting a sufficiently strong expander into two edge-disjoint spanning expanders.

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