Lecture 17: Approximation algorithm for max-cut

Given an undirected graph G = (V,E), the max-cut problem asks for the partition S1, S2 ⊆ V , s.t., the number of edges going from S1 to S2 are maximized. Remember that since S1, S2 is a partition, so S1∪S2 = V and S1 ∩ S2 = ∅. Clearly this is an optimization problem. It is known to be NP-hard and so not expected to have a polynomial time algorithm. We are interested in finding an approximation algorithm for this problem. Many problems in the real world can be formulated as a max-cut instance. For example, suppose the graph gives information about the friendships in a set of students. More concretely, the vertices correspond to students in the class and there is an edge present between two students if they are friends. The problem is to put them in two different classroom so that the number of friendships (edges) across the two rooms is maximized (one measure to stop copying). This is essentially solving a max-cut problem on the given graph.