RL for Combinatorial Optimization

The Job Shop Scheduling Problem (JSSP) is commonly formulated as a disjunctive graph in which nodes represent operations and edges encode technological precedence constraints as well as machine-sharing conflicts. Most existing reinforcement learning approaches model this graph as homogeneous, merging job-precedence and machine-contention edges into a single relation type. Such a simplification overlooks the intrinsic heterogeneity of the problem structure and may lead to the loss of critical relational information. To address this limitation, we propose the Heterogeneous Graph Transformer (HGT)-Scheduler, a reinforcement learning framework that models the JSSP as a heterogeneous graph. The proposed architecture leverages a Heterogeneous Graph Transformer to capture type-specific relational patterns through edge-type-dependent attention mechanisms applied to precedence and contention relations. The scheduling policy is trained using Proximal Policy Optimization. The effectiveness of the proposed method is evaluated on the Fisher--Thompson benchmark instances. On the FT06 instance, the HGT-Scheduler achieves an optimality gap of 8.4%, statistically outperforming both an identical architecture that ignores edge types ($p = 0.011$) and a standard Graph Isomorphism Network baseline. On the larger FT10 instance, the approach demonstrates favorable scalability. However, under a 50,000-step training limit, the performance of heterogeneous and homogeneous graph models is comparable, suggesting that edge-type awareness requires longer training horizons for larger problem instances. Ablation analyses further indicate that a three-layer attention architecture provides the best performance. Overall, the results confirm that explicitly modeling distinct edge semantics improves the learning of effective scheduling policies.

Multi-objective reinforcement learning (MORL) is effective for multi-echelon combinatorial supply chain optimisation, where tasks involve high dimensionality, uncertainty, and competing objectives. However, its deployment in dynamic environments is hindered by the need for task-specific retraining and substantial computational cost. We introduce MIRACL (Meta multI-objective Reinforcement leArning with Composite Learning), a hierarchical Meta-MORL framework that allows for a few-shot generalisation across diverse tasks. MIRACL decomposes each task into structured subproblems for efficient policy adaptation and meta-learns a global policy across tasks using a Pareto-based adaptation strategy to encourage diversity in meta-training and fine-tuning. To our knowledge, this is the first integration of Meta-MORL with such mechanisms in combinatorial optimisation. Although validated in the supply chain domain, MIRACL is theoretically domain-agnostic and applicable to broader dynamic multi-objective decision-making problems. Empirical evaluations show that MIRACL outperforms conventional MORL baselines in simple to moderate tasks, achieving up to 10% higher hypervolume and 5% better expected utility. These results underscore the potential of MIRACL for robust, efficient adaptation in multi-objective problems.

Railcar shunting is a core planning task in freight railyards, where yard planners need to disassemble and reassemble groups of railcars to form outbound trains. Classification tracks with access from one side only can be considered as stack structures, where railcars are added and removed from only one end, leading to a last-in-first-out (LIFO) retrieval order. In contrast, two-sided tracks function like queue structures, allowing railcars to be added from one end and removed from the opposite end, following a first-in-first-out (FIFO) order. We consider a problem requiring assembly of multiple outbound trains using two locomotives in a railyard with two-sided classification track access. To address this combinatorially challenging problem class, we decompose the problem into two subproblems, each with one-sided classification track access and a locomotive on each side. We present a novel Hybrid Heuristic-Reinforcement Learning (HHRL) framework that integrates railway-specific heuristic solution approaches with a reinforcement learning method, specifically Q-learning. The proposed framework leverages methods to decrease the state-action space and guide exploration during reinforcement learning. The results of a series of numerical experiments demonstrate the efficiency and quality of the HHRL algorithm in both one-sided access, single-locomotive problems and two-sided access, two-locomotive problems.

The rise of smart manufacturing under Industry 4.0 introduces mass customization and dynamic production, demanding more advanced and flexible scheduling techniques. The flexible job-shop scheduling problem (FJSP) has attracted significant attention due to its complex constraints and strong alignment with real-world production scenarios. Current deep reinforcement learning (DRL)-based approaches to FJSP predominantly employ constructive methods. While effective, they often fall short of reaching (near-)optimal solutions. In contrast, improvement-based methods iteratively explore the neighborhood of initial solutions and are more effective in approaching optimality. However, the flexible machine allocation in FJSP poses significant challenges to the application of this framework, including accurate state representation, effective policy learning, and efficient search strategies. To address these challenges, this paper proposes a Memory-enhanced Improvement Search framework with heterogeneous graph representation--MIStar. It employs a novel heterogeneous disjunctive graph that explicitly models the operation sequences on machines to accurately represent scheduling solutions. Moreover, a memoryenhanced heterogeneous graph neural network (MHGNN) is designed for feature extraction, leveraging historical trajectories to enhance the decision-making capability of the policy network. Finally, a parallel greedy search strategy is adopted to explore the solution space, enabling superior solutions with fewer iterations. Extensive experiments on synthetic data and public benchmarks demonstrate that MIStar significantly outperforms both traditional handcrafted improvement heuristics and state-of-the-art DRL-based constructive methods.

Multi-task Vehicle Routing Problems (VRPs) aim to minimize routing costs while satisfying diverse constraints. Existing solvers typically adopt a unified reinforcement learning (RL) framework to learn generalizable patterns across tasks. However, they often overlook the constraint and node dynamics during the decision process, making the model fail to accurately react to the current context. To address this limitation, we propose Chain-of-Context Learning (CCL), a novel framework that progressively captures the evolving context to guide fine-grained node adaptation. Specifically, CCL constructs step-wise contextual information via a Relevance-Guided Context Reformulation (RGCR) module, which adaptively prioritizes salient constraints. This context then guides node updates through a Trajectory-Shared Node Re-embedding (TSNR) module, which aggregates shared node features from all trajectories' contexts and uses them to update inputs for the next step. By modeling evolving preferences of the RL agent, CCL captures step-by-step dependencies in sequential decision-making. We evaluate CCL on 48 diverse VRP variants, including 16 in-distribution and 32 out-of-distribution (with unseen constraints) tasks. Experimental results show that CCL performs favorably against the state-of-the-art baselines, achieving the best performance on all in-distribution tasks and the majority of out-of-distribution tasks.

The Flexible Job Shop Scheduling Problem (FJSP) originates from real production lines, while some practical constraints are often ignored or idealized in current FJSP studies, among which the limited buffer problem has a particular impact on production efficiency. To this end, we study an extended problem that is closer to practical scenarios--the Flexible Job Shop Scheduling Problem with Limited Buffers and Material Kitting. In recent years, deep reinforcement learning (DRL) has demonstrated considerable potential in scheduling tasks. However, its capacity for state modeling remains limited when handling complex dependencies and long-term constraints. To address this, we leverage a heterogeneous graph network within the DRL framework to model the global state. By constructing efficient message passing among machines, operations, and buffers, the network focuses on avoiding decisions that may cause frequent pallet changes during long-sequence scheduling, thereby helping improve buffer utilization and overall decision quality. Experimental results on both synthetic and real production line datasets show that the proposed method outperforms traditional heuristics and advanced DRL methods in terms of makespan and pallet changes, and also achieves a good balance between solution quality and computational cost. Furthermore, a supplementary video is provided to showcase a simulation system that effectively visualizes the progression of the production line.

Winter road maintenance is critical for ensuring public safety and reducing environmental impacts, yet existing methods struggle to manage large-scale routing problems effectively and mostly reply on human decision. This study presents a novel, scalable bi-level optimization framework, validated on real operational data on UK strategic road networks (M25, M6, A1), including interconnected local road networks in surrounding areas for vehicle traversing, as part of the highway operator's efforts to solve existing planning challenges. At the upper level, a reinforcement learning (RL) agent strategically partitions the road network into manageable clusters and optimally allocates resources from multiple depots. At the lower level, a multi-objective vehicle routing problem (VRP) is solved within each cluster, minimizing the maximum vehicle travel time and total carbon emissions. Unlike existing approaches, our method handles large-scale, real-world networks efficiently, explicitly incorporating vehicle-specific constraints, depot capacities, and road segment requirements. Results demonstrate significant improvements, including balanced workloads, reduced maximum travel times below the targeted two-hour threshold, lower emissions, and substantial cost savings. This study illustrates how advanced AI-driven bi-level optimization can directly enhance operational decision-making in real-world transportation and logistics.

This paper introduces and formalizes the Dynamic and Stochastic Vehicle Routing Problem with Emission Quota (DS-QVRP-RR), a novel routing problems that integrates dynamic demand acceptance and routing with a global emission constraint. A key contribution is a two-layer optimization framework designed to facilitate anticipatory rejections of demands and generation of new routes. To solve this, we develop hybrid algorithms that combine reinforcement learning with combinatorial optimization techniques. We present a comprehensive computational study that compares our approach against traditional methods. Our findings demonstrate the relevance of our approach for different types of inputs, even when the horizon of the problem is uncertain.

The Multiple Traveling Salesman Problem (mTSP) extends the Traveling Salesman Problem to m tours that start and end at a common depot and jointly visit all customers exactly once. In the min-max variant, the objective is to minimize the longest tour, reflecting workload balance. We propose a hybrid approach, Construct, Merge, Solve & Adapt with Reinforcement Learning (RL-CMSA), for the symmetric single-depot min-max mTSP. The method iteratively constructs diverse solutions using probabilistic clustering guided by learned pairwise q-values, merges routes into a compact pool, solves a restricted set-covering MILP, and refines solutions via inter-route remove, shift, and swap moves. The q-values are updated by reinforcing city-pair co-occurrences in high-quality solutions, while the pool is adapted through ageing and pruning. This combination of exact optimization and reinforcement-guided construction balances exploration and exploitation. Computational results on random and TSPLIB instances show that RL-CMSA consistently finds (near-)best solutions and outperforms a state-of-the-art hybrid genetic algorithm under comparable time limits, especially as instance size and the number of salesmen increase.

Automated placement of components on printed circuit boards (PCBs) is a critical stage in placement layout design. While reinforcement learning (RL) has been successfully applied to system-on-chip IP block placement and chiplet arrangement in complex packages, PCB component placement presents unique challenges due to several factors: variation in component sizes, single- and double-sided boards, wirelength constraints, board constraints, and non-overlapping placement requirements. In this work, we adopt a component-centric layout for automating PCB component placement using RL: first, the main component is fixed at the center, while passive components are placed in proximity to the pins of the main component. Free space around the main component is discretized, drastically reducing the search space while still covering all feasible placement; second, we leverage prior knowledge that each passive's position has to be near to its corresponding voltage source. This allows us to design the reward function which avoids wasted exploration of infeasible or irrelevant search space. Using the component centric layout, we implemented different methods including Deep Q-Network, Actor-Critic algorithm and Simulated Annealing. Evaluation on over nine real-world PCBs of varying complexity shows that our best proposed method approaches near human-like placements in terms of wirelength and feasibility.

Reinforcement learning (RL) is a subfield of machine learning that focuses on developing models that can autonomously learn optimal decision-making strategies over time. In a recent pioneering paper, Wagner demonstrated how the Deep Cross-Entropy RL method can be applied to tackle various problems from extremal graph theory by reformulating them as combinatorial optimization problems. Subsequently, many researchers became interested in refining and extending the framework introduced by Wagner, thereby creating various RL environments specialized for graph theory. Moreover, a number of problems from extremal graph theory were solved through the use of RL. In particular, several inequalities concerning the Laplacian spectral radius of graphs were refuted, new lower bounds were obtained for certain Ramsey numbers, and contributions were made to the Turán-type extremal problem in which the forbidden structures are cycles of length three and four. Here, we present Reinforcement Learning for Graph Theory (RLGT), a novel RL framework that systematizes the previous work and provides support for both undirected and directed graphs, with or without loops, and with an arbitrary number of edge colors. The framework efficiently represents graphs and aims to facilitate future RL-based research in extremal graph theory through optimized computational performance and a clean and modular design.

Emergency situations that require the evacuation of urban areas can arise from man-made causes (e.g., terrorist attacks or industrial accidents) or natural disasters, the latter becoming more frequent due to climate change. As a result, effective and fast methods to develop evacuation plans are of great importance. In this work, we identify and propose the Bus Evacuation Orienteering Problem (BEOP), an NP-hard combinatorial optimization problem with the goal of evacuating as many people from an affected area by bus in a short, predefined amount of time. The purpose of bus-based evacuation is to reduce congestion and disorder that arises in purely car-focused evacuation scenarios. To solve the BEOP, we propose a deep reinforcement learning-based method utilizing graph learning, which, once trained, achieves fast inference speed and is able to create evacuation routes in fractions of seconds. We can bound the gap of our evacuation plans using an MILP formulation. To validate our method, we create evacuation scenarios for San Francisco using real-world road networks and travel times. We show that we achieve near-optimal solution quality and are further able to investigate how many evacuation vehicles are necessary to achieve certain bus-based evacuation quotas given a predefined evacuation time while keeping run time adequate.

Neural algorithmic reasoning (NAR) is a paradigm that trains neural networks to execute classic algorithms by supervised learning. Despite its successes, important limitations remain: inability to construct valid solutions without post-processing and to reason about multiple correct ones, poor performance on combinatorial NP-hard problems, and inapplicability to problems for which strong algorithms are not yet known. To address these limitations, we reframe the problem of learning algorithm trajectories as a Markov decision process, which imposes structure on the solution construction procedure and unlocks the powerful tools of imitation and reinforcement learning (RL). We propose the GNARL framework, encompassing the methodology to translate problem formulations from NAR to RL and a learning architecture suitable for a wide range of graph-based problems. We achieve high rates of reaching correct states on several CLRS-30 problems and performance matching or exceeding much narrower NAR approaches for NP-hard problems. Remarkably, GNARL remains applicable when no expert algorithm is available, though reward-driven learning can exhibit greater variability than direct supervision.

Pipe routing is a highly complex, time-consuming, and no-deterministic polynomial-time hard (NP-hard) problem in aeroengine design. Despite extensive research efforts in optimizing constant-curvature pipe routing, the growing demand for free-form pipes poses new challenges. Dynamic design environments and fuzzy layout rules further impact the optimization performance and efficiency. To tackle these challenges, this study proposes a self-learning-based method (SLPR) for optimizing free-form pipe routing in aeroengines. The SLPR is based on the proximal policy optimization (PPO) algorithm and integrates a unified rule modeling framework for efficient obstacle detection and fuzzy rule modeling in continuous space. Additionally, a potential energy table is constructed to enable rapid queries of layout tendencies and interference. The agent within SLPR iteratively refines pipe routing and accumulates the design knowledge through interaction with the environment. Once the design environment shifts, the agent can swiftly adapt by fine-tuning network parameters. Comparative tests reveal that SLPR ensures smooth pipe routing through cubic non-uniform B-spline (NURBS) curves, avoiding redundant pipe segments found in constant-curvature pipe routing. Results in both static and dynamic design environments demonstrate that SLPR outperforms three representative baselines in terms of the pipe length reduction, the adherence to layout rules, the path complexity, and the computational efficiency. Furthermore, tests in dynamic environments indicate that SLPR eliminates labor-intensive searches from scratch and even yields superior solutions compared to the retrained model. These results highlight the practical value of SLPR for real-world pipe routing, meeting lightweight, precision, and sustainability requirements of the modern aeroengine design.

Due to the ongoing electrification of transportation systems, energy-based routing problems such as the Electric Vehicle Routing Problem (EVRP) are becoming increasingly important. Unlike Euclidean routing problems, the EVRP involves asymmetric energy costs caused by road topology. Existing learning-based routing approaches are based on graph neural networks (GNNs) or transformer architectures operating on Euclidean node coordinates, making them poorly suited for asymmetric, non-Euclidean edge-based metrics such as energy consumption. To address this, we propose the Graph Edge Attention Network (GREAT), a novel GNN-related architecture that operates purely in edge space rather than on node coordinates, making it naturally compatible with asymmetric routing costs. Integrated into an end-to-end reinforcement learning framework, GREAT jointly optimizes routing and charging decisions for the EVRP. We train the model using real-world energy consumption data from San Francisco, derived from physical road networks enriched with topological features, and evaluate generalization on datasets from San Francisco, Vienna, and Gothenburg. Across diverse battery sizes (15-40 kWh) and customer demand distributions, GREAT outperforms a strong HGS-based EVRP heuristic in most tested settings, achieving energy reductions of up to 8.78% while reducing runtime by a factor of 60. Moreover, we benchmark against a custom adaptive large neighborhood search (ALNS) that jointly optimizes routing and charging. GREAT achieves competitive performance at a faster runtime, and we show that the ALNS can profit from a warm start with the GREAT solution. We further evaluate GREAT on standard (asymmetric) routing problems, including Traveling Salesman Problem, Capacitated Vehicle Routing Problem, and Orienteering Problem, where it achieves state-of-the-art performance among neural methods for non-Euclidean settings.

Deep neural networks based on reinforcement learning (RL) for solving combinatorial optimization (CO) problems are developing rapidly and have shown a tendency to approach or even outperform traditional solvers. However, existing methods overlook an important distinction: CO problems differ from other traditional problems in that they focus solely on the optimal solution provided by the model within a specific length of time, rather than considering the overall quality of all solutions generated by the model. In this paper, we propose Leader Reward and apply it during two different training phases of the Policy Optimization with Multiple Optima (POMO) model to enhance the model's ability to generate optimal solutions. This approach is applicable to a variety of CO problems, such as the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), and the Flexible Flow Shop Problem (FFSP), but also works well with other POMO-based models or inference phase's strategies. We demonstrate that Leader Reward greatly improves the quality of the optimal solutions generated by the model. Specifically, we reduce the POMO's gap to the optimum by more than 100 times on TSP100 with almost no additional computational overhead.