Concerning Iterative Graph Normalization and Maximum Weight Independent Sets

We consider a very simple dynamical system on weighted graphs which we call Iterative Graph Normalization (IGN) and a variant in which we apply a non-linear activation function to the weights after each normalization. We show that the indicator vectors of the Maximal Independent Sets of the graph are the only binary fixed points of IGN, that they are attractive under simple conditions on the activation function and we characterize their basins of attraction. We enumerate a number of other fixed points and we prove repulsivity for some classes. Based on extensive experiments and different theoretical arguments we conjecture that IGN always converges and converges to a binary solution for non-linear activations. If our conjectures are correct, IGN would thus be a differentiable approximation algorithm for the Maximum Weight Independent Set problem (MWIS), a central NP-hard optimization problem with numerous applications. IGN is closely related to a greedy approximation algorithm of MWIS by Kako et al. which has a proven approximation ratio. Experimental results show that IGN provides solutions of very similar quality. In the context of the Assignment Problem, IGN corresponds to an iterative matrix normalization scheme which is closely related to the Sinkhorn-Knopp algorithm except that it projects to a permutation matrix instead of a doubly stochastic matrix. We relate our scheme to the Softassign algorithm and provide comparative results. As Graph Normalization is differentiable, its iterations can be embedded into a machine learning framework and used to train end-to-end any model which includes a graphical optimization step which can be cast as a maximum weight independent set problem. This includes problems such as graph and hypergraph matching, sequence alignment, clustering, ranking, etc. with applications in multiple domains.