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

TL;DR

FFCG introduces a novel reinforcement-learning framework for Column Generation that selects a variable number of columns per iteration by modeling column selection as an MDP and using a polynomial-time, sequential column-selection strategy. The reward design accounts for both objective improvement and redundancy penalties, and a marginal Q-value approximator is trained with experience replay to guide decisions. Empirically, FFCG achieves substantial speedups on CSP and VRPTW benchmarks, reducing CG iterations by up to 84% and computing time by up to 84% compared with strong baselines. The approach demonstrates robust generalization across problem sizes and highlights the benefit of dynamic, data-driven column selection over fixed or greedy strategies, with practical implications for solving large-scale LP relaxations efficiently.

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.

Figures (7)

• Figure 1: Column Selection in FFCG: First, the available action set is $\mathcal{G}=\{v_4,v_5,v_6,v_7\}$ (green node), and the RL agent selects $v_4$ (the selected node is orange). Then, a new available action $\mathrm{STOP}$ (blue node), which means stop column selection and return the selected columns, is added to action space $\mathcal{G}-\{v_4\}$. The RL agent repeatedly updates the state and selects from the remaining actions until the action $\mathrm{STOP}$ is selected. Finally, the suggested columns set $\mathcal{C}=\{v_4,v_6,v_7\}$ is returned.
• Figure 2: CSP: CG convergence plots for FFCG (green), Greedy-S (red), MLCG-S (blue), RLCG-S (gray), Greedy-M (orange), and RLMCS-M (purple). The solid curves are the mean of the objective values for all instances, and the shaded area shows the standard deviation.
• Figure 3: CSP, $n=50$ test instances: The suggested column size per CG iteration for FFCG, Greedy-M, and RLMCS-M.
• Figure 4: CSP, $n=200$ test instances: The suggested column size per CG iteration for FFCG, Greedy-M, and RLMCS-M.
• Figure 5: CSP, $n=750$ test instances: The suggested column size per CG iteration for FFCG, Greedy-M, and RLMCS-M.