A nondifferentiable optimization approach to ratio-cut partitioning

We propose a new method for finding the minimum ratio-cut of a graph. Ratio-cut is NP-hard problem for which the best previously known algorithm gives an O(log n)-factor approximation by solving its dually related maximum concurrent flow problem. We formulate the minimum ratio-cut as a certain nondifferentiable optimization problem, and show that the global minimum of the optimization problem is equal to the minimum ratio-cut. Moreover, we provide strong symbolic computation based evidence that any strict local minimum gives an approximation by a factor of 2. We also give an efficient heuristic algorithm for finding a local minimum of the proposed optimization problem based on standard nondifferentiable optimization methods and evaluate its performance on several families of graphs. We achieve O(n^1.6) experimentally obtained running time on these graphs.