Physics-inspired graph neural networks (PI-GNNs) have been utilized as an efficient unsupervised framework for relaxing combinatorial optimization problems encoded through a specific graph structure and loss, reflecting dependencies between the problem's variables. While the framework has yielded promising results in various combinatorial problems, we show that the performance of PI-GNNs systematically plummets with an increasing density of the combinatorial problem graphs. Our analysis reveals an interesting phase transition in the PI-GNNs' training dynamics, associated with degenerate solutions for the denser problems, highlighting a discrepancy between the relaxed, real-valued model outputs and the binary-valued problem solutions. To address the discrepancy, we propose principled alternatives to the naive strategy used in PI-GNNs by building on insights from fuzzy logic and binarized neural networks. Our experiments demonstrate that the portfolio of proposed methods significantly improves the performance of PI-GNNs in increasingly dense settings.

Combinatorial optimization (CO) problems are crucial in various scientific and industrial applications. Recently, researchers have proposed using unsupervised Graph Neural Networks (GNNs) to address NP-hard combinatorial optimization problems, which can be reformulated as Quadratic Unconstrained Binary Optimization (QUBO) problems. GNNs have demonstrated high performance with nearly linear scalability and significantly outperformed classic heuristic-based algorithms in terms of computational efficiency on large-scale problems. However, when utilizing standard node features, GNNs tend to get trapped to suboptimal local minima of the energy landscape, resulting in low quality solutions. We introduce a novel algorithm, denoted hereafter as QRF-GNN, leveraging the power of GNNs to efficiently solve CO problems with QUBO formulation. It relies on unsupervised learning by minimizing the loss function derived from QUBO relaxation. The proposed key components of the architecture include the recurrent use of intermediate GNN predictions, parallel convolutional layers and combination of static node features as input. Altogether, it helps to adapt the intermediate solution candidate to minimize QUBO-based loss function, taking into account not only static graph features, but also intermediate predictions treated as dynamic, i.e. iteratively changing recurrent features. The performance of the proposed algorithm has been evaluated on the canonical benchmark datasets for maximum cut, graph coloring and maximum independent set problems. Results of experiments show that QRF-GNN drastically surpasses existing learning-based approaches and is comparable to the state-of-the-art conventional heuristics, improving their scalability on large instances.

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 real-world graph learning tasks require handling dynamic graphs where new nodes and edges emerge. Dynamic graph learning methods commonly suffer from the catastrophic forgetting problem, where knowledge learned for previous graphs is overwritten by updates for new graphs. To alleviate the problem, continual graph learning methods are proposed. However, existing continual graph learning methods aim to learn new patterns and maintain old ones with the same set of parameters of fixed size, and thus face a fundamental tradeoff between both goals. In this paper, we propose Parameter Isolation GNN (PI-GNN) for continual learning on dynamic graphs that circumvents the tradeoff via parameter isolation and expansion. Our motivation lies in that different parameters contribute to learning different graph patterns. Based on the idea, we expand model parameters to continually learn emerging graph patterns. Meanwhile, to effectively preserve knowledge for unaffected patterns, we find parameters that correspond to them via optimization and freeze them to prevent them from being rewritten. Experiments on eight real-world datasets corroborate the effectiveness of PI-GNN compared to state-of-the-art baselines.

Teaching Graph Neural Networks (GNNs) to accurately classify nodes under severely noisy labels is an important problem in real-world graph learning applications, but is currently underexplored. Although pairwise training methods have demonstrated promise in supervised metric learning and unsupervised contrastive learning, they remain less studied on noisy graphs, where the structural pairwise interactions (PI) between nodes are abundant and thus might benefit label noise learning rather than the pointwise methods. This paper bridges the gap by proposing a pairwise framework for noisy node classification on graphs, which relies on the PI as a primary learning proxy in addition to the pointwise learning from the noisy node class labels. Our proposed framework PI-GNN contributes two novel components: (1) a confidence-aware PI estimation model that adaptively estimates the PI labels, which are defined as whether the two nodes share the same node labels, and (2) a decoupled training approach that leverages the estimated PI labels to regularize a node classification model for robust node classification. Extensive experiments on different datasets and GNN architectures demonstrate the effectiveness of PI-GNN, yielding a promising improvement over the state-of-the-art methods. Code is publicly available at https://github.com/TianBian95/pi-gnn.

AWS Center for Quantum Computing, Pasadena, CA 91125, USA (Dated: March 23, 2023) We provide a comprehensive reply to the comment written by Stefan Boettcher [arXiv:2210.00623] Conversely, we highlight the broader algorithmic development underlying our original work [1], and (within our original framework) provide additional numerical results showing sizable improvements over our original data, thereby refuting the comment's original performance statements. Furthermore, it has already been shown that physics-inspired graph neural networks (PI-GNNs) can outperform greedy algorithms, in particular on hard, dense instances. We also argue that the internal (parallel) anatomy of graph neural networks is very different from the (sequential) nature of greedy algorithms, and (based on their usage at the scale of real-world social networks) point out that graph neural networks have demonstrated their potential for superior scalability compared to existing heuristics such as extremal optimization. Finally, we conclude highlighting the conceptual novelty of our work and outline some potential extensions.