Graph Normalization: Fast Binarizing Dynamics for Differentiable MWIS

2026-05-06

AI Summary

This work addresses the NP-hard maximum weight independent set (MWIS) problem by proposing a Graph Normalization (GN) dynamical system that integrates quasi-Newton methods with majorization-minimization strategies to efficiently produce differentiable hard-decision solutions. Theoretically, it establishes the first convergence guarantee of GN to binary indicator vectors of maximal independent sets, generalizes the Motzkin–Straus theorem, and reveals an equivalence between GN and replicator dynamics. Algorithmically, it naturally extends Sinkhorn iterations to hard assignments on arbitrary constraint graphs. Empirical results demonstrate that the method achieves high-quality solutions within 1% of optimality on real-world graphs with millions of edges in just seconds on a CPU, significantly advancing the efficiency and scalability of differentiable combinatorial optimization.

Abstract

We introduce Graph Normalization (GN), a principled dynamical system on graphs that serves as a differentiable approximation engine for the NP-hard Maximum Weight Independent Set (MWIS) problem. MWIS encompasses many combinatorial challenges, including optimal assignment, scheduling, set packing, and MAP inference in discrete Markov Random Fields. Unlike Belief Propagation, we prove GN always converges to a binary indicator of a Maximum Independent Set. GN realizes a fast quasi-Newton descent through an exact Majorization-Minimization step, systematically improving the MWIS relaxed primal objective. We establish an equivalence between GN and the Replicator Dynamics of a nonlinear evolutionary game, where vertices compete for inclusion in an independent set. While a non-potential game, the GN game follows Fisher's Fundamental Theorem of Natural Selection, where the average fitness equals the MWIS primal objective and strictly increases. This connection leads to a weighted extension of the Motzkin-Straus theorem, showing MISes are in bijection with the local minima of a quadratic form over a tilted simplex. For the Assignment Problem, GN acts as a variant of the Sinkhorn algorithm that naturally converges to a hard assignment while generalizing to arbitrary constraint graphs. We demonstrate GN's performance as a fast binarization engine for the state-of-the-art Bregman-Sinkhorn relaxed MWIS solver. On real-world benchmarks with up to 1M edges, GN identifies solutions within 1% of the best known results in seconds on a CPU. GN opens new avenues for deep learning architectures requiring differentiable, "hard" decisions under constraints, with applications in structured sparse attention, dynamic network pruning, and Mixture-of-Experts. Beyond core AI, the GN framework enables end-to-end learning of constrained optimization in computer vision, computational biology, and resource allocation.

Problem

• Maximum Weight Independent Set
• Combinatorial Optimization
• NP-hard Problem
• Graph Constraints
• Discrete Optimization

Innovation

• Graph Normalization
• Maximum Weight Independent Set
• Differentiable Optimization
• Replicator Dynamics
• Majorization-Minimization