We give the first combinatorial approximation algorithm for Maxcut that beats the trivial 0.5 factor by a constant. The main partitioning procedure is very intuitive, natural, and easily described. It essentially performs a number of random walks and aggregates the information to provide the partition. We can control the running time to get an approximation factor-running time tradeoff. We show that for any constant b > 1.5, there is an O(n^b)algorithm that outputs a (0.5+d) approximation for Maxcut, where d = d(b) is some positive constant. One of the components of our algorithm is a weak local graph partitioning procedure that may be of independent interest. Given a starting vertex i and a conductance parameter f, unless a random walk of length l = O(log n) starting from i mixes rapidly (in terms of f and l), we can find a cut of conductance at most f close to the vertex. The work done per vertex found in the cut is sublinear in n. This is joint work with C. Seshadhri.
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