Enhancing Column Generation by a Machine-Learning-Based Pricing Heuristic for Graph Coloring

Abstract

Column Generation (CG) is an effective method for solving large-scale optimization problems. CG starts by solving a subproblem with a subset of columns (i.e., variables) and gradually includes new columns that can improve the solution of the current subproblem. The new columns are generated as needed by repeatedly solving a pricing problem, which is often NP-hard and is a bottleneck of the CG approach. To tackle this, we propose a Machine-Learning-based Pricing Heuristic (MLPH) that can generate many high-quality columns efficiently. In each iteration of CG, our MLPH leverages an ML model to predict the optimal solution of the pricing problem, which is then used to guide a sampling method to efficiently generate multiple high-quality columns. Using the graph coloring problem, we empirically show that MLPH significantly enhances CG as compared to six state-of-the-art methods, and the improvement in CG can lead to substantially better performance of the branch-and-price exact method. Machine Learning Based Pricing HeuristicGiven the MWISP at a CG iteration, we employ an ML model to predict which vertices belong to the optimal MIS. This prediction is then used to guide a sampling method to generate high-quality MISs efficiently. Having many MISs, we introduce several strategies to select a subset of these to form the new RMP at the next CG iteration.