Column Generation (CG) is an iterative algorithm for solving linear programs (LPs) with an extremely large number of variables (columns). CG is the workhorse for tackling large-scale integer linear programs, which rely on CG to solve LP relaxations within a branch and bound algorithm. Two canonical applications are the Cutting Stock Problem (CSP) and Vehicle Routing Problem with Time Windows (VRPTW). In VRPTW, for example, each binary variable represents the decision to include or exclude a route, of which there are exponentially many; CG incrementally grows the subset of columns being used, ultimately converging to an optimal solution. We propose RLCG, the first Reinforcement Learning (RL) approach for CG. Unlike typical column selection rules which myopically select a column based on local information at each iteration, we treat CG as a sequential decision-making problem, as the column selected in an iteration affects subsequent iterations of the algorithm. This perspective lends itself to a Deep Reinforcement Learning approach that uses Graph Neural Networks (GNNs) to represent the variable-constraint structure in the LP of interest. We perform an extensive set of experiments using the publicly available BPPLIB benchmark for CSP and Solomon benchmark for VRPTW. RLCG converges faster and reduces the number of CG iterations by 22.4% for CSP and 40.9% for VRPTW on average compared to a commonly used greedy policy.

This paper considers the learning of Boolean rules in either disjunctive normal form (DNF, OR-of-ANDs, equivalent to decision rule sets) or conjunctive normal form (CNF, AND-of-ORs) as an interpretable model for classification. An integer program is formulated to optimally trade classification accuracy for rule simplicity. Column generation (CG) is used to efficiently search over an exponential number of candidate clauses (conjunctions or disjunctions) without the need for heuristic rule mining. This approach also bounds the gap between the selected rule set and the best possible rule set on the training data. To handle large datasets, we propose an approximate CG algorithm using randomization. Compared to three recently proposed alternatives, the CG algorithm dominates the accuracy-simplicity trade-off in 8 out of 16 datasets. When maximized for accuracy, CG is competitive with rule learners designed for this purpose, sometimes finding significantly simpler solutions that are no less accurate.

Neural Information Processing Systems http://nips.cc/

Interpretability has become a well-recognized goal for machine learning models.

Column generation and branch-and-price are leading methods for large-scale exact optimization. Column generation iterates between solving a master problem and a pricing problem. The master problem is a linear program, which can be solved using a generic solver. The pricing problem is highly dependent on the application but is usually discrete. Due to the difficulty of discrete optimization, high-performance column generation often relies on a custom pricing algorithm built specifically to exploit the problem's structure. This bespoke nature of the pricing solver prevents the reuse of components for other applications. We show that domain-independent dynamic programming, a software package for modeling and solving arbitrary dynamic programs, can be used as a generic pricing solver. We develop basic implementations of branch-and-price with pricing by domain-independent dynamic programming and show that they outperform a world-leading solver on static mixed integer programming formulations for seven problem classes.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper proposes a fairer optimization criterion, "regularized maximin", for centralized multi-agent MDPs. The idea, taken from the networking literature is elegant. The authors also propose an iterative optimization method that scales somewhat better than linear programming. The description of the transition model, lines 69-79, seems unnecessarily detailed.

The team formation and routing problem is a challenging optimization problem with several real-world applications in fields such as airport, healthcare, and maintenance operations. To solve this problem, exact solution methods based on column generation have been proposed in the literature. In this paper, we propose a novel partial column generation strategy for settings with multiple pricing problems, based on predicting which ones are likely to yield columns with a negative reduced cost. We develop a machine learning model tailored to the team formation and routing problem that leverages graph neural networks for these predictions. Computational experiments demonstrate that applying our strategy enhances the solution method and outperforms traditional partial column generation approaches from the literature, particularly on hard instances solved under a tight time limit.

Abstract--Metro crew planning is a key component of smart city development as it directly impacts the operational efficiency and service reliability of public transportation. With the rapid expansion of metro networks, effective multi-line scheduling and emergency management have become essential for large-scale seamless operations. However, current research focuses primarily on individual metro lines, with insufficient attention on cross-line coordination and rapid replanning during disruptions. Here, a unified optimization framework is presented for multi-line metro crew planning and replanning with heterogeneous workforce. Specifically, a hierarchical time-space network model is proposed to represent the unified crew action space, and computationally efficient constraints and formulations are derived for the crew's heterogeneous qualifications and preferences. Solution algorithms based on column generation and shortest path adjustment are further developed, utilizing the proposed network model. Experiments with real data from Shanghai and Beijing Metro demonstrate that the proposed methods outperform benchmark heuristics in both cost reduction and task completion, and achieve notable efficiency gains by incorporating cross-line operations, particularly for urgent tasks during disruptions. This work highlights the role of global optimization and cross-line coordination in multi-line metro system operations, providing insights into the efficient and reliable functioning of public transportation in smart cities. Metro systems are vital to urban transportation, offering high efficiency and large capacity to meet growing mobility demands. Within the context of metro operations, labor costs account for a significant share of expenses [1]. Consequently, metro crew planning plays a crucial factor in achieving smooth, cost-effective operations. As metro systems continue to expand rapidly, the need for optimized crew planning approaches has become increasingly critical to realize efficient and intelligent metro operations that support the broader goals of smart city development [2]. Existing research on metro crew planning primarily focuses on single-line operations [3], [4], [5], [6], [7], [8].