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 + δ)-approximation for MaxCut, where δ = δ(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 φ, unless a random walk of length = O(log n) starting from i mixes rapidly (in terms of φ and ), we can find a cut of conductance at most φ close to the vertex. The work done per vertex found in the cut is sublinear in n.
In REF Kale and Seshadhri present a combinatorial approximation algorithm for Max-Cut using random walks, which gives a (0.5 + δ)-approximation, where δ is some positive constant which appears also in the running time of the algorithm.
Combinatorial Approximation Algorithms for MaxCut using Random Walks
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