Graph Algorithm Based Submodular Function for Sparsest Cut Problem

Sparsest cut problems are very important graph partitions, which have been widely applied in expander graphs, Markov chains, and image segmentation. In this paper, we study the edge-weighted version of the Sparse Cut Problem, which minimizes the ratio of the total weight of edges between blocks and the total weight of edges incident to vertices in one block. We first prove that the problem is even NP-hard for an edge-weighted graph with bridges. Then, we combine and generalize submodular functions and principal partition to design a graph algorithm to improve the initial bipartition, which runs in polynomial time by using network flow as its subroutines.