Combinatorial Optimization with Graph Convolutional Networks and Guided Tree Search (1810.10659v1)

Abstract: We present a learning-based approach to computing solutions for certain NP-hard problems. Our approach combines deep learning techniques with useful algorithmic elements from classic heuristics. The central component is a graph convolutional network that is trained to estimate the likelihood, for each vertex in a graph, of whether this vertex is part of the optimal solution. The network is designed and trained to synthesize a diverse set of solutions, which enables rapid exploration of the solution space via tree search. The presented approach is evaluated on four canonical NP-hard problems and five datasets, which include benchmark satisfiability problems and real social network graphs with up to a hundred thousand nodes. Experimental results demonstrate that the presented approach substantially outperforms recent deep learning work, and performs on par with highly optimized state-of-the-art heuristic solvers for some NP-hard problems. Experiments indicate that our approach generalizes across datasets, and scales to graphs that are orders of magnitude larger than those used during training.

Combinatorial Optimization with Graph Convolutional Networks and Guided Tree Search

The paper "Combinatorial Optimization with Graph Convolutional Networks and Guided Tree Search" addresses the challenge of solving NP-hard problems using an innovative combination of deep learning and traditional heuristic algorithms. Specifically, the approach leverages Graph Convolutional Networks (GCNs) to approximate solutions for complex combinatorial problems that are otherwise computationally intractable to solve optimally.

Overview of the Approach

The central technique in this work involves training a GCN to estimate the likelihood of a vertex in a graph being part of the optimal solution for a given NP-hard problem. This GCN-based method integrates the strengths of deep learning with established heuristic strategies to enable effective exploration of large solution spaces. By constructing a diverse set of probability maps through supervised learning, the GCN can guide a parallel tree search process, thus generating a multitude of candidate solutions swiftly.

Evaluation and Empirical Results

The presented method was evaluated on canonical NP-hard problems, including Satisfiability (SAT), Maximal Independent Set (MIS), Minimum Vertex Cover (MVC), and Maximal Clique (MC). The evaluation also spanned across different datasets such as synthetic SATLIB benchmarks and real-world social network graphs, which encompass up to a hundred thousand nodes.

Significant findings from the empirical experiments include:

• Performance Evaluation: The proposed approach consistently outperformed recent deep learning-based methods for NP-hard problems, such as those based on reinforcement learning. Notably, on the SATLIB dataset, the GCN combined with tree search solved all test instances, a feat unmatched by other contemporary deep learning approaches.
• Comparison with Heuristic Solvers: Performance was on par with, or exceeded, that of state-of-the-art heuristic solvers when applied to various NP-hard problems, highlighting the efficacy of the GCN-guided tree search.
• Scalability: The approach demonstrated significant scalability and generalization capabilities. It was successfully applied to graph instances substantially larger than those on which the network was trained, suggesting robust pattern recognition in diverse graph structures.

Implications and Future Directions

The practical implications of this research are noteworthy, as it suggests a promising avenue for leveraging machine learning, particularly GCNs, in the domain of combinatorial optimization. The demonstrated ability to generalize across datasets and problem scales indicates that this method could be adapted to solve other NP-hard problems beyond what was addressed directly, such as by training networks specifically for problems like Maximal Clique or TSP.

This work opens several potential directions for future research. One area could involve exploring the application of similar deep learning architectures to other classes of NP-hard problems, potentially delivering efficient approximations or heuristic solutions more generally. Additionally, investigating the integration of reinforcement learning with GCNs in this framework could further enhance solution quality and search efficiency.

Conclusion

This research highlights an innovative integration of GCNs and heuristic algorithms in addressing NP-hard problems, with implications that extend well beyond the immediate findings. It showcases the potential of learning-based approaches as powerful tools in the field of combinatorial optimization, emphasizing adaptability and generalization across different problem instances, datasets, and graph scales. As machine learning continues to advance, approaches like this one may become integral in crafting scalable solutions to a wide array of computationally challenging problems.