Combinatorial approximation algorithms for MAXCUT using random walks.

Abstract

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 {tilde O}(n{sup 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 {ell} = O(log n) starting from i mixes rapidly (in terms of {phi} and {ell}), 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.