Solution stability in linear programming relaxations: graph partitioning and unsupervised learning

We propose a new method to quantify the solution stability of a large class of combinatorial optimization problems arising in machine learning. As practical example we apply the method to correlation clustering, clustering aggregation, modularity clustering, and relative performance significance clustering. Our method is extensively motivated by the idea of linear programming relaxations. We prove that when a relaxation is used to solve the original clustering problem, then the solution stability calculated by our method is conservative, that is, it never overestimates the solution stability of the true, unrelaxed problem. We also demonstrate how our method can be used to compute the entire path of optimal solutions as the optimization problem is increasingly perturbed. Experimentally, our method is shown to perform well on a number of benchmark problems.