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 $(0.5+\delta)$ approximation for Maxcut, where $\delta = \delta(b)$ 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 $\phi$, unless a random walk of length $l = O(\log n)$ starting from $i$ mixes rapidly (in terms of $\phi$ and $l$), we can find a cut of conductance at most $\phi$ close to the vertex. The work done per vertex found in the cut is sublinear in $n$.