In recent years, the advantages of Graph Neural Networks (GNNs) in solving complex combinatorial optimization problems have become more and more obvious. Under this background, this paper demonstrates a method based on GNNs to solve large-scale Unconstrained 0–1 Quadratic Programming (UBQP) problems. Firstly, the UBQP problem models of different combinatorial optimization problems are converted into a unified graph model. This approach makes up for the shortcoming that GNNs cannot be effectively applied to combinatorial optimization problems not based on graph. Secondly, based on Graph Attention Network (GAT) algorithm, this study designs a general GNN model capable of solving UBQP problems. In addition, a random sampling scheme for the complete graph structure is devised, which greatly alleviates the pronounced over-smoothing phenomenon when the model is applied to complete graphs and speeds up the convergence rate. In this paper, a supervised learning approach is adopted to train and test a unified structured model (without changing network parameters) on UBQP problems of varying sizes, and to evaluate the model's generalization ability on high-dimensional UBQP problems. Furthermore, an unsupervised learning mechanism is utilized to train and solve individual sample problems of different dimensions. These experimental results consistently demonstrate that the model and strategy presented in this paper exhibit strong solving ability and generalization ability for combinatorial optimization problems.
In recent years, the advantages of Graph Neural Networks (GNNs) in solving complex combinatorial optimization problems have become more and more obvious. Under this background, this paper demonstrates a method based on GNNs to solve large-scale Unconstrained 0–1 Quadratic Programming (UBQP) problems. Firstly, the UBQP problem models of different combinatorial optimization problems are converted into a unified graph model. This approach makes up for the shortcoming that GNNs cannot be effectively applied to combinatorial optimization problems not based on graph. Secondly, based on Graph Attention Network (GAT) algorithm, this study designs a general GNN model capable of solving UBQP problems. In addition, a random sampling scheme for the complete graph structure is devised, which greatly alleviates the pronounced over-smoothing phenomenon when the model is applied to complete graphs and speeds up the convergence rate. In this paper, a supervised learning approach is adopted to train and test a unified structured model (without changing network parameters) on UBQP problems of varying sizes, and to evaluate the model's generalization ability on high-dimensional UBQP problems. Furthermore, an unsupervised learning mechanism is utilized to train and solve individual sample problems of different dimensions. These experimental results consistently demonstrate that the model and strategy presented in this paper exhibit strong solving ability and generalization ability for combinatorial optimization problems.
In recent years, there has been notable interest in investigating combinatorial optimization (CO) problems by neural-based framework. An emerging strategy to tackle these challenging problems involves the adoption of graph neural networks (GNNs) as an alternative to traditional algorithms, a subject that has attracted considerable attention. Despite the growing popularity of GNNs and traditional algorithm solvers in the realm of CO, there is limited research on their integrated use and the correlation between them within an end-to-end framework. The primary focus of our work is to formulate a more efficient and precise framework for CO by employing decision-focused learning on graphs. Additionally, we introduce a decision-focused framework that utilizes GNNs to address CO problems with auxiliary support. To realize an end-to-end approach, we have designed two cascaded modules: (a) an unsupervised trained graph predictive model, and (b) a solver for quadratic binary unconstrained optimization. Empirical evaluations are conducted on various classical tasks, including maximum cut, maximum independent set, and minimum vertex cover. The experimental results on classical CO problems (i.e. MaxCut, MIS, and MVC) demonstrate the superiority of our method over both the standalone GNN approach and classical methods.
The logistics vehicle routing optimization problem is a widely discussed combinatorial optimization(CO) problem with broad application prospects. Graph Neural Network(GNN) is extensively used in this context due to their suitability for the graph structure of transportation networks. However, existing methods lack interpretability for CO problem models and do not offer a generalized and unified modeling process. Additionally, traditional neural networks face shortcomings when handling dynamic graphs and directed graph problems, with suboptimal node information aggregation capabilities. Therefore, designing a method for the logistics vehicle routing optimization problem that features a unified and concise problem model, rapid solution speed, and accurate results is crucial. This paper proposes a framework for accelerating the solution of the logistics vehicle routing optimization problem using a Graph Attention Network (GAT) based on quadratic unconstrained binary op-timization(QUBO) modeling methods. The framework aims to abstract practical problems into QUBO form, and it constructs a concise and unified model for the problem while utilizing Multi-Head Attention mechanisms to aggregate node information in graph-structured problems. These are then solved using GNN. Comparing this method with actual results and traditional GNN solving methods demonstrates that, in cases with a higher number of nodes, this method has certain advantages in solution speed and accuracy.
We investigate a link between Graph Neural Networks (GNNs) and Quadratic Unconstrained Binary Optimization (QUBO) problems, laying the groundwork for GNNs to approximate solutions for these computationally challenging tasks. By analyzing the sensitivity of QUBO formulations, we frame the solution of QUBO problems as a heterophilic node classification task. We then propose QUBO-GNN, an architecture that integrates graph representation learning techniques with QUBO-aware features to approximate solutions efficiently. Additionally, we introduce a self-supervised data generation mechanism to enable efficient and scalable training data acquisition even for large-scale QUBO instances. Experimental evaluations of QUBO-GNN across diverse QUBO problem sizes demonstrate its superior performance compared to exhaustive search and heuristic methods. Finally, we discuss open challenges in the emerging intersection between QUBO optimization and GNN-based learning.
In recent years, graph neural networks (GNNs) have become increasingly popular for solving NP-hard combinatorial optimization (CO) problems, such as maximum cut and maximum independent set. The core idea behind these methods is to represent a CO problem as a graph and then use GNNs to learn the node/graph embedding with combinatorial information. Although these methods have achieved promising results, given a specific CO problem, the design of GNN architectures still requires heavy manual work with domain knowledge. Existing automated GNNs are mostly focused on traditional graph learning problems, which is inapplicable to solving NP-hard CO problems. To this end, we present a new class of \textbf{AUTO}mated \textbf{G}NNs for solving \textbf{NP}-hard problems, namely \textbf{AutoGNP}. We represent CO problems by GNNs and focus on two specific problems, i.e., mixed integer linear programming and quadratic unconstrained binary optimization. The idea of AutoGNP is to use graph neural architecture search algorithms to automatically find the best GNNs for a given NP-hard combinatorial optimization problem. Compared with existing graph neural architecture search algorithms, AutoGNP utilizes two-hop operators in the architecture search space. Moreover, AutoGNP utilizes simulated annealing and a strict early stopping policy to avoid local optimal solutions. Empirical results on benchmark combinatorial problems demonstrate the superiority of our proposed model.
Quadratic Unconstrained Binary Optimization (QUBO) is a generic technique to model various NP-hard Combinatorial Optimization problems (CO) in the form of binary variables. Ising Hamiltonian is used to model the energy function of a system. QUBO to Ising Hamiltonian is regarded as a technique to solve various canonical optimization problems through quantum optimization algorithms. Recently, PI-GNN, a generic framework, has been proposed to address CO problems over graphs based on Graph Neural Network (GNN) architecture. They introduced a generic QUBO-formulated Hamiltonian-inspired loss function that was directly optimized using GNN. PI-GNN is highly scalable but there lies a noticeable decrease in the number of satisfied constraints when compared to problem-specific algorithms and becomes more pronounced with increased graph densities. Here, We identify a behavioral pattern related to it and devise strategies to improve its performance. Another group of literature uses Reinforcement learning (RL) to solve the aforementioned NP-hard problems using problem-specific reward functions. In this work, we also focus on creating a bridge between the RL-based solutions and the QUBO-formulated Hamiltonian. We formulate and empirically evaluate the compatibility of the QUBO-formulated Hamiltonian as the generic reward function in the RL-based paradigm in the form of rewards. Furthermore, we also introduce a novel Monty Carlo Tree Search-based strategy with GNN where we apply a guided search through manual perturbation of node labels during training. We empirically evaluated our methods and observed up to 44% improvement in the number of constraint violations compared to the PI-GNN.
Many complex problems encountered in both production and daily life can be conceptualized as combinatorial optimization problems (COPs). Many ad-hoc deep learning methods have been proposed to solve these problems, but there still lacks an effective unified framework. In this work, we take a step towards this goal by designing an unified end-to-end deep reinforcement learning framework named CONQUER. CONQUER first translates various COPs into the unified QUBO (Quadratic Unconstrained Binary Optimization) formalization, and afterwards trains a neural QUBO solver to search high-quality strategies, which are then checked and repaired to obtain the solutions for original COPs. The QUBO solver is a purely data-driven neural model, which does not rely on any expert inputs. We adopt the graph transformer network to represent the QUBO graph, so as to effectively capture the features of problem objective and constraints. The Deep Q-network is utilized to train the QUBO solver to find long-sighted solution strategies. Despite being trained solely on small synthetic graphs, CONQUER exhibits promising generality on much larger instances, and can be easily adapted to real-world datasets. Experimentally, we show that CONQUER achieves the state-of-the-art performances on four COPs over graphs, including minimum vertex cover, maximum independent set, maximum clique and maximum cut problem.