We describe a new approximation algorithm for Max Cut. Our algorithm runs inÕ(n 2 ) time, where n is the number of vertices, and achieves an approximation ratio of .531. On instances in which an optimal solution cuts a 1 − ε fraction of edges, our algorithm finds a solution that cuts a 1−4 √ ε+ Our main result is a variant of spectral partitioning, which can be implemented in nearly linear time. Given a graph in which the Max Cut optimum is a 1 − ε fraction of edges, our spectral partitioning algorithm finds a set S of vertices and a bipartition L, R = S − L of S such that at least a 1 − O( √ ε) fraction of the edges incident on S have one endpoint in L and one endpoint in R. (This can be seen as an analog of Cheeger's inequality for the smallest eigenvalue of the adjacency matrix of a graph.) Iterating this procedure yields the approximation results stated above. A different, more complicated, variant of spectral partitioning leads to a polynomial time algorithm that cuts a 1/2 + e −Ω(1/ε) fraction of edges in graphs in which the optimum is 1/2 + ε.
Trevisan REF proposed an approximate algorithm for max-cut by the smallest eigenvalue with approximation ratio of 0.531.
Max cut and the smallest eigenvalue
STOC '09
Mathematics Computer Science