Short Paths in Expander Graphs

Graph expansion has proved to be a powerful general tool for analyzing the behavior of routing algorithms and the inter{connection networks on which they run. We develop new routing algorithms and structural results for bounded{degree expander graphs. Our results are uniied by the fact that they are all based upon, and extend, a body of work asserting that expanders are rich in short, disjoint paths. In particular , our work has consequences for the disjoint paths problem, multicommodity ow, and graph minor containment. We show: (i) A greedy algorithm for approximating the maximum disjoint paths problem achieves a polylogarithmic approximation ratio in bounded{degree expanders. Although our algorithm is both deterministic and on-line, its performance guarantee is an improvement over previous bounds in expanders. (ii) For a multicommodity ow problem with arbitrary demands on a bounded{degree expander, there is a (1 + "){optimal solution using only ow paths of polylogarithmic length. It follows that the multicom-modity ow algorithm of Awerbuch and Leighton runs in nearly linear time per commodity in expanders. Our analysis is based on establishing the following: given edge weights on an expander G, one can increase some of the weights very slightly so the resulting shortest-path metric is smooth { the min-weight path between any pair of nodes uses a polylogarithmic number of edges. (iii) Every bounded{degree expander on n nodes contains every graph with O(n= log O(1) n) nodes and edges as a minor.