CS261: A Second Course in Algorithms Lecture #20: The Maximum Cut Problem and Semidefinite Programming∗

Now that you’re finishing CS261, you’re well equipped to comprehend a lot of advanced material on algorithms. This lecture illustrates this point by teaching you about a cool and famous approximation algorithm. In the maximum cut problem, the input is an undirected graph G = (V,E) with a nonnegative weight we ≥ 0 for each edge e ∈ E. The goal is to compute a cut — a partition of the vertex set into sets A and B — that maximizes the total weight of the cut edges (the edges with one endpoint in each of A and B). Now, if it were the minimum cut problem, we’d know what to do — that problem reduces to the maximum flow problem (Exercise Set #2). It’s tempting to think that we can reduce the maximum cut problem to the minimum cut problem just by negating the weights of all of the edges. Such a reduction would yield a minimum cut problem with negative weights (or capacities). But if you look back at our polynomial-time algorithms for computing minimum cuts, you’ll notice that we assumed nonnegative edge capacities, and that our proofs depended on this assumption. Indeed, it’s not hard to prove that the maximum cut problem is NP -hard. So, let’s talk about polynomial-time approximation algorithms. It’s easy to come up with a 1 2 -approximation algorithm for the maximum cut problem. Almost anything works — a greedy algorithm, local search, picking a random cut, linear programming rounding, and so on. But frustratingly, none of these techniques seemed capable of proving an approximation factor better than 1 2 . This made it remarkable when, in 1994, Goemans and Williamson showed how a new technique, “semidefinite programming rounding,” could be used to blow away all previous approximation algorithms for the maximum cut problem.