Cooperative Cuts: Graph Cuts with Submodular Edge Weights

We introduce a problem we call Cooperative cut, where the goal is to find a minimum-cost graph cut but where a submodular function is used to define the cost of a subsets of edges. That means, the cost of an edge that is added to the current cut set C depends on the edges in C. This generalization of 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 problem. On the positive side, we propose and compare four approximation algorithms with an overall approximation factor of min { |V |/2, |C∗|, O( √ |E| log |V |), |Pmax| } , where C∗ is the optimal solution, and Pmax is the longest s, t path across the cut between given s, t. We also introduce additional heuristics for the problem which have attractive properties from the perspective of practical applications and implementations in that existing fast min-cut libraries may be used as subroutines. Both our approximation algorithms, and our heuristics, appear to do well in practice.