Deep k-grouping: An Unsupervised Learning Framework for Combinatorial Optimization on Graphs and Hypergraphs

TL;DR

Deep k-grouping presents an unsupervised learning framework for k-grouping combinatorial optimization on graphs and hypergraphs by formulating problems as unconstrained OH-QUBO/OH-PUBO objectives and solving them with a GPU-accelerated differentiable optimizer. The approach unifies graph and hypergraph coloring and partitioning within a single objective-design, and employs a Gini coefficient-based annealing strategy to promote discreteness while avoiding local optima. Empirical results show that the method outperforms traditional solvers such as SCIP and Tabu on large-scale instances and provides competitive performance against neural baselines, with clear scalability advantages. The work introduces a practical, scalable pathway for challenging CO tasks in domains like VLSI design, image segmentation, and distributed computing.

Abstract

Along with AI computing shining in scientific discovery, its potential in the combinatorial optimization (CO) domain has also emerged in recent years. Yet, existing unsupervised neural network solvers struggle to solve $k$-grouping problems (e.g., coloring, partitioning) on large-scale graphs and hypergraphs, due to limited computational frameworks. In this work, we propose Deep $k$-grouping, an unsupervised learning-based CO framework. Specifically, we contribute: Novel one-hot encoded polynomial unconstrained binary optimization (OH-PUBO), a formulation for modeling k-grouping problems on graphs and hypergraphs (e.g., graph/hypergraph coloring and partitioning); GPU-accelerated algorithms for large-scale k-grouping CO problems. Deep $k$-grouping employs the relaxation of large-scale OH-PUBO objectives as differentiable loss functions and trains to optimize them in an unsupervised manner. To ensure scalability, it leverages GPU-accelerated algorithms to unify the training pipeline; A Gini coefficient-based continuous relaxation annealing strategy to enforce discreteness of solutions while preventing convergence to local optima. Experimental results demonstrate that Deep $k$-grouping outperforms existing neural network solvers and classical heuristics such as SCIP and Tabu.

Figures (7)

• Figure 1: Overview of unsupervised neural network-based CO frameworks, including QUBO-based neural network solvers and the proposed Deep $k$-grouping framework.
• Figure 2: Comparison of the runtime between SCIP and Deep $k$-grouping.
• Figure 3: Quality of solutions of the max-cut problem with or without Gini coefficient-based annealing strategy across various graph sizes.
• Figure 4: Quality and discreteness of solutions of the max-cut problem with or without Gini coefficient-based annealing strategy.
• Figure 5: Quality and discreteness of solutions of the graph partitioning problem with or without Gini coefficient-based annealing strategy.