Globally optimizing graph partitioning problems using message passing

Graph partitioning algorithms play a central role in data analysis and machine learning. Most useful graph partitioning criteria correspond to optimizing a ratio between the cut and the size of the partitions, this ratio leads to an NP-hard problem that is only solved approximately. This makes it difficult to know whether failures of the algorithm are due to failures of the optimization or to the criterion being optimized. In this paper we present a framework that seeks and finds the optimal solution of several NP-hard graph partitioning problems. We use a classical approach to ratio problems where we repeatedly ask whether the optimal solution is greater than or less than some constant - λ. Our main insight is the equivalence between this "λ question" and performing inference in a graphical model with many local potentials and one high-order potential. We show that this specific form of the highorder potential is amenable to message-passing algorithms and how to obtain a bound on the optimal solution from the messages. Our experiments show that in many cases our approach yields the global optimum and improves the popular spectral solution.