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Edge disjoint paths in moderately connected graphs
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Given an expander graph $G=(V,E)$ and a set of $q$ disjoint pairs of vertices in $V$, the authors are interested in finding for each pair $(a_i, b_i)$ a path connecting $a_i$ to $b_i$ such that the set of $q$ paths so found is edge disjoint. (For general graphs the related decision problem is NP complete.) The authors prove sufficient conditions for the existence of edge-disjoint paths connecting any set of $q\leq n/(\log n)^\kappa$ disjoint pairs of vertices on any $n$ vertex bounded degree expander, where $\kappa$ depends only on the expansion properties of the input graph, and not on $n$. Furthermore, a randomized $o(n^3)$ time algorithm, and a random $\cal NC$ algorithm for constructing these paths is presented. (Previous existence proofs and construction algorithms allowed only up to $n^\epsilon$ pairs, for some $\epsilon\ll \frac{1}{3}$, and strong expanders [D. Peleg and E. Upfal, Combinatorica, 9 (1989), pp.~289--313.].) In passing, an algorithm is developed for splitting a sufficiently strong expander into two edge-disjoint spanning expanders.