Fine-Grain GPU Parallelization of the Generalized Partition Crossover for Large-Scale Traveling Salesman Problems

Evolutionary multitasking allows several related problems to be solved in a single run of an algorithm. In this paper, we investigate integrating evolutionary multitasking with Edge Assembly Crossover (MT-EAX) to solve the classical Travelling Salesperson Problem (TSP). To fairly compare MT-EAX against standard EAX under strict compute budgets, we evaluate three scaling methods: generation scaling, population scaling, and balanced scaling. Our results show that generationally scaled MT-EAX is highly effective compute-wise in the early stages of the search, saving $60\%$ to $90\%$ of compute for equal or better solution quality. We observe that instance geometry has a significant impact, with clustered, normally distributed instances securing larger improvements than uniformly distributed ones. However, when scaling by population or utilising explicit solution transfer, the results are negative due to population starvation and incompatible cross-instance parent selection. We demonstrate that the advantage of MT-EAX derives from increased diversity through parallel search in early generations, which can be successfully preserved using a decoupled configuration to often strictly outperform or match standard EAX performance at final convergence.

Solving large-scale instances of the Traveling Salesman Problem (TSP) exactly is computationally expensive. Researchers often employ graph sparsification methods to improve computational efficiency. Traditional sparsification methods typically rely on fixed heuristics and fail to fully exploit instance-specific structural information. In this paper, we propose Graph Edge Sparsification (GES), a learning-based sparsification approach for Euclidean TSP. By incorporating geometric structural information and combinatorial optimization technology, our proposed method adaptively generates a sparsification graph for different instances, significantly reducing the graph size and accelerating the solving process. Experimental results demonstrate that our sparsification method can prune up to 95% of edges on the MATILDA dataset, while keeping the solution gap within 1% of the optimal value. Moreover, our approach exhibits strong generalization capability on the TSPLIB benchmark.In some large-scale instances, the pruning rate exceeds 99%, while the optimality gap remains below 1%.

The maximum independent set (MIS) problem is a fundamental NP-hard combinatorial optimization problem with applications in scheduling, resource allocation, and network analysis. Exact solvers can provide high-quality solutions or optimality certificates, but their computational cost grows rapidly with graph size, while hand-crafted heuristics improve scalability at the expense of guarantees. Learning-based methods offer an alternative by exploiting structural patterns across graph instances, yet directly predicting independent sets can make global coordination difficult on large graphs. We instead use learning to guide multilevel graph coarsening while retaining combinatorial search for final decision making. Our Dual-GNN Multilevel Coarsening framework uses a Partition GNN to score candidate contractions and a Representative GNN to select top-k local independent-set states for each final cluster. Experiments on Erdős--Rényi graphs with up to 2,000 vertices demonstrate a favorable quality--runtime trade-off. On 500-vertex instances with certified optima, our method achieves an average independent-set size of 19.20, corresponding to 99.5% of the optimal value of 19.30, while reducing the mean wall-clock time from 643.57 seconds for exact solving to 3.41 seconds, yielding an approximately 189$\times$ speedup. On larger graphs with 1,000 and 2,000 vertices, our method achieves the best mean solution quality among all evaluated methods. Moreover, although trained only on Erdős--Rényi graphs with edge probability $p=0.35$, the learned coarsening policy generalizes effectively across both unseen graph densities and structurally different graph families.