3 Problem Definition

Given an undirected edge-weighted graph, G = (V,E), the maximum cut problem (Max-Cut) is to find a bipartition of the vertices that maximizes the weight of the edges crossing the partition. If the edge weights are non-negative, then this problem is equivalent to finding a maximum weight subset of the edges that forms a bipartite subgraph, i.e. the maximum bipartite subgraph problem. All results discussed in this article assume non-negative edge weights. Max-Cut is one of Karp’s original NP-complete problems [Kar72]. In fact, it is NP-hard to approximate to within a factor better than 16 17 [TSSW00, H̊as01]. For nearly twenty years, the best-known approximation factor for Max-Cut was half, which can be achieved by a very simple algorithm: Form a set S by placing each vertex in S with probability half. Since each edge crosses the cut (S, V \S) with probability half, the expected value of this cut is half the total edge weight. This implies that for any graph, there exists a cut with value at least half of the total edge weight. In 1976, Sahni and Gonzalez presented a deterministic half-approximation algorithm for Max-Cut, which is essentially a de-randomization of the aforementioned randomized algorithm [SG76]: Iterate through the vertices and form sets S and S̄ by placing each vertex in the set that maximizes the weight of cut (S, S̄) thus far. After each iteration of this process, the weight of this cut will be at least half of the weight of the edges with both endpoints in S ∪ S̄. This simple half-approximation algorithm uses the fact that for any graph with non-negative edge weights, the total edge weight of a given graph is an upper bound on the value of its maximum cut. There exist classes of graphs for which a maximum cut is arbitrarily close to half the total edge weight, i.e. graphs for which this “trivial” upper bound can be close to twice the true value of an optimal solution. An example of such a class of graphs are complete graphs on n vertices, Kn.