Balanced Graph Partitioning is a classical optimization problem where quality guarantees are computationally infeasible, and practical solvers therefore rely on manually engineered heuristics. Yet, the problem has also proven difficult for approaches that rely heavily on machine learning - especially since applications often need to partition graphs of huge scale in a short amount of time. Instead, we demonstrate how to achieve practical improvements with a more careful approach that uses machine learning to improve heuristic decisions within the state-of-the-art solver Mt-KaHyPar. We use a pre-trained neural network to predict a score for each edge, which then guides clustering decisions in the first phase of the partitioning (the coarsening). Combined with corresponding adjustments to the clustering algorithm and an efficient implementation of the neural network logic, we improve the overall solution quality while preserving the efficiency and scalability of the original algorithm. Our detailed evaluation on more than 180 graphs shows an average quality improvement of 2% on a class of graphs with beneficial properties, and unchanged quality on all remaining graphs. Moreover, our improvements generalize to a set of instances from the literature that are much larger than the graphs used during training.

The graph coloring problem asks for an assignment of the minimum number of distinct colors to vertices in an undirected graph with the constraint that no pair of adjacent vertices share the same color. The problem is a thoroughly studied NP-hard combinatorial problem with several real-world applications. As such, a number of greedy heuristics have been suggested that strike a good balance between coloring quality, execution time, and also parallel scalability. In this work, we introduce a graph neural network (GNN) based ordering heuristic and demonstrate that it outperforms existing greedy ordering heuristics both on quality and performance. Previous results have demonstrated that GNNs can produce high-quality colorings but at the expense of excessive running time. The current paper is the first that brings the execution time down to compete with existing greedy heuristics. Our GNN model is trained using both supervised and unsupervised techniques. The experimental results show that a 2-layer GNN model can achieve execution times between the largest degree first (LF) and smallest degree last (SL) ordering heuristics while outperforming both on coloring quality. Increasing the number of layers improves the coloring quality further, and it is only at four layers that SL becomes faster than the GNN. Finally, our GNN-based coloring heuristic achieves superior scaling in the parallel setting compared to both SL and LF.

Learn and Reduce Dataset This dataset consists of vertex-labeled graphs used to train machine learning models for data reduction screening. This is used as a preprocessing step to solve the Maximum Weight Independent Set problem. The original folder contains all instances and reduction data using the computationally cheap reduction rules. The graphs and reduction data are stored as two separate CSV files, with the suffix _original_graph.csv and _original_reduction_data.csv. Similarly, the kernel folder contains the non-empty reduced instances. Here, we use the suffix _kernel_graph.csv and _kernel_reduction_data.csv. The graph files have two columns, source and target. Each undirected edge {u,v} in the graph appears as both u;v and v;u.

Identifying a maximum independent set is a fundamental NP-hard problem. This problem has several real-world applications and requires finding the largest possible set of vertices not adjacent to each other in an undirected graph. Over the past few years, branch-and-bound and branch-and-reduce algorithms have emerged as some of the most effective methods for solving the problem exactly. Specifically, the branch-and-reduce approach, which combines branch-and-bound principles with reduction rules, has proven particularly successful in tackling previously unmanageable real-world instances. This progress was largely made possible by the development of more effective reduction rules. Nevertheless, other key components that can impact the efficiency of these algorithms have not received the same level of interest. Among these is the branching strategy, which determines which vertex to branch on next. Until recently, the most widely used strategy was to choose the vertex of the highest degree. In this work, we present a graph neural network approach for selecting the next branching vertex. The intricate nature of current branch-and-bound solvers makes supervised and reinforcement learning difficult. Therefore, we use a population-based genetic algorithm to evolve the model’s parameters instead. Our proposed approach results in a speedup on 73% of the benchmark instances with a median speedup of 24%.

Minimum weighted vertex cover is the NP-hard graph problem of choosing a subset of vertices incident to all edges such that the sum of the weights of the chosen vertices is minimum. Previous efforts for solving this in practice have typically been based on search-based iterative heuristics or exact algorithms that rely on reduction rules and branching techniques. Although exact methods have shown success in solving instances with up to millions of vertices efficiently, they are limited in practice due to the NP-hardness of the problem. We present a new hybrid method that combines elements from exact methods, iterative search, and graph neural networks (GNNs). More specifically, we first compute a greedy solution using reduction rules whenever possible. If no such rule applies, we consult a GNN model that selects a vertex that is likely to be in or out of the solution, potentially opening up for further reductions. Finally, we use an improved local search strategy to enhance the solution further. Extensive experiments on graphs of up to a billion edges show that the proposed GNN-based approach finds better solutions than existing heuristics. Compared to exact solvers, the method produced solutions that are, on average, 0.04% away from the optimum while taking less time than all state-of-the-art alternatives.