Towards Solving Polynomial-Objective Integer Programming with Hypergraph Neural Networks

2026-03-16

AI Summary

This work addresses polynomial integer programming problems involving high-order variable interactions—a class of optimization challenges significantly more difficult than linear integer programs due to their nonlinear structure. The paper introduces the first solution framework based on hypergraph neural networks, which constructs a high-order-term-aware hypergraph representation to uniformly capture complex dependencies among variables, high-order terms, and constraints. A dual-path hypergraph convolution mechanism is designed to separately aggregate variable–high-order-term and variable–constraint information for predicting high-quality initial solutions, subsequently refined via heuristic search. Experiments demonstrate that the proposed method substantially outperforms existing learning-based approaches and commercial solvers across multiple benchmarks, achieving notable advances in both solution quality and computational efficiency, and is applicable to general polynomial integer programming scenarios.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Programming
• Search and Optimization: Heuristic Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

Complex real-world optimization problems often involve both discrete decisions and nonlinear relationships between variables. Many such problems can be modeled as polynomial-objective integer programs, encompassing cases with quadratic and higher-degree variable interactions. Nonlinearity makes them more challenging than their linear counterparts. In this paper, we propose a hypergraph neural network (HNN) based method to solve polynomial-objective integer programming (POIP). Besides presenting a high-degree-term-aware hypergraph representation to capture both high-degree information and variable-constraint interdependencies, we also propose a hypergraph neural network, which integrates convolution between variables and high-degree terms alongside convolution between variables and constraints, to predict solution values. Finally, a search process initialized from the predicted solutions is performed to further refine the results. Comprehensive experiments across a range of benchmarks demonstrate that our method consistently outperforms both existing learning-based approaches and state-of-the-art solvers, delivering superior solution quality with favorable efficiency. Note that our experiments involve both polynomial objectives and constraints, demonstrating our HNN's versatility for general POIP problems and highlighting its advancement over the existing literature.

Problem

• polynomial-objective integer programming
• nonlinear optimization
• discrete optimization
• hypergraph representation
• integer programming

Innovation

• hypergraph neural network
• polynomial-objective integer programming
• high-degree term modeling
• variable-constraint interdependency
• learning-based optimization