FFCG: Effective and Fast Family Column Generation for Solving Large-Scale Linear Program

TL;DR

This paper introduces FFCG, a reinforcement learning approach for column generation in large-scale linear programs, significantly improving convergence speed and reducing computational time by adaptively selecting multiple columns per iteration.

Contribution

The paper proposes a novel RL-based method for column selection in CG, addressing the state-space explosion problem and outperforming existing approaches in efficiency.

Findings

77.1% fewer iterations for CSP

84.8% fewer iterations for VRPTW

71.4% reduction in computing time for CSP

Abstract

Column Generation (CG) is an effective and iterative algorithm to solve large-scale linear programs (LP). During each CG iteration, new columns are added to improve the solution of the LP. Typically, CG greedily selects one column with the most negative reduced cost, which can be improved by adding more columns at once. However, selecting all columns with negative reduced costs would lead to the addition of redundant columns that do not improve the objective value. Therefore, selecting the appropriate columns to add is still an open problem and previous machine-learning-based approaches for CG only add a constant quantity of columns per iteration due to the state-space explosion problem. To address this, we propose Fast Family Column Generation (FFCG) -- a novel reinforcement-learning-based CG that selects a variable number of columns as needed in an iteration. Specifically, we formulate the column selection problem in CG as an MDP and design a reward metric that balances both the convergence speed and the number of redundant columns. In our experiments, FFCG converges faster on the common benchmarks and reduces the number of CG iterations by 77.1% for Cutting Stock Problem (CSP) and 84.8% for Vehicle Routing Problem with Time Windows (VRPTW), and a 71.4% reduction in computing time for CSP and 84.0% for VRPTW on average compared to several state-of-the-art baselines.