The Applicability and Scalability of Graph Neural Networks on Combinatorial Optimization

Abstract: This master's thesis investigates the application of Graph Neural Networks (GNNs) to address scalability challenges in combinatorial optimization, with a primary focus on the minimum Total Dominating set Problem (TDP) and additionally the related Carrier Scheduling Problem (CSP) in networks of Internet of Things. The research identifies the NP-hard nature of these problems as a fundamental challenge and addresses how to improve predictions on input graphs of sizes much larger than seen during training phase. Further, the thesis explores the instability in such scalability when leveraging GNNs for TDP and CSP. Two primary measures to counter this scalability problem are proposed and tested: incorporating node degree as an additional feature and modifying the attention mechanism in GNNs. Results indicate that these countermeasures show promise in addressing scalability issues in TDP, with node degree inclusion demonstrating overall performance improvements while the modified attention mechanism presents a nuanced outcome with some metrics improved at the cost of others. Application of these methods to CSP yields bleak results, evincing the challenges of scalability in more complex problem domains. The thesis contributes by detecting and addressing scalability challenges in combinatorial optimization using GNNs and provides insights for further research in refining methodologies for real-world applications.