Notes on graph cuts with submodular edge weights

Generalizing the cost in the standard min-cut problem to a submodular cost function immediately makes the problem harder. Not only do we prove NP hardness even for nonnegative submodular costs, but also show a lower bound of Ω(|V |) on the approximation factor for the (s, t) cut version of the problem. On the positive side, we propose and compare three approximation algorithms with an overall approximation factor of O(min{|V |, √ |E| log |V |}) that appear to do well in practice.