A Max-Cut approximation using a graph based MBO scheme

Abstract

The Max-Cut problem is a well known combinatorial optimization problem. In this paper we describe a fast approximation method. Given a graph G, we want to find a cut whose size is maximal among all possible cuts. A cut is a partition of the vertex set of G into two disjoint subsets. For an unweighted graph, the size of the cut is the number of edges that have one vertex on either side of the partition; we also consider a weighted version of the problem where each edge contributes a nonnegative weight to the cut.We introduce the signless Ginzburg-Landau functional and prove that this functional Γ-converges to a Max-Cut objective functional. We approximately minimize this functional using a graph based signless Merriman-Bence-Osher scheme, which uses a signless Laplacian. We show experimentally that on some classes of graphs the resulting algorithm produces more accurate maximum cut approximations than the current state-of-the-art approximation algorithm. One of our methods of minimizing the functional results in an algorithm with a time complexity of O(|E|), where |E| is the total number of edges on G.

Partial differential equations and variational methods for geometric processing of images

“…The success of the Ginzburg-Landau clustering technique to approximate the NP hard balanced clustering problem suggests that other similarly hard combinatorial problems might be approachable using these techniques. In [52] the solution to the max-cut problem (i.e. partition the node set of a graph into two sets such that the cut between them is maximal) is approximated by minimizers of the graph functional…”

Graph MBO on Star Graphs and Regular Trees. With Corrections to DOI 10.1007/s00032-014-0216-8

“…In recent years the graph version of this process and variations thereof have been succesfully applied to data clustering and classifcation problems and other graph based problems, e.g. in [4,5,8,9,10,7,15,3], which in turn has prompted further theoretical study of the MBO scheme on graphs [14,1].…”