Recovery in Perturbation - Stable Maximum Cut Instances ∗

Today we continue our discussion of exact recovery under perturbation-stability assumptions. We’ll look at a different cut problem than last lecture, and this will also give us an excuse to touch on two cool and fundamental topics, metric embeddings and semidefinite programming. The Maximum Cut problem is a famous NP -hard problem. The input is an undirected graph G = (V,E), where each edge e ∈ E has a positive weight we > 0. The goal is to compute a cut (A,B) that maximizes the weight of the crossing edges (i.e., the edges with one endpoint in each of A,B). The Maximum Cut problem can arise in applications as a type of 2-clustering problem. For example, if weights represent dissimilarities (between images, genomes, etc.), then it makes sense to define clusters so that pairs of objects in different clusters tend to be quite dissimilar. The fact that the Maximum Cut problem is NP -hard may be confusing, given that the Minimum s-t Cut problem is polynomial-time solvable (as proved last lecture or by a reduction to the maximum flow problem). 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 an instance of Minimum Cut with negative weights (or capacities). But all of the polynomial-time algorithms for solving the Minimum Cut problem require nonnegative edge capacities. Indeed, it’s not hard to prove that the maximum cut problem is NP -hard. The goal for today, analogous to last lecture, is to recover the optimal solution in polynomial time in γ-perturbation-stable instances, where γ is as small as possible. The definition