Title: Hybrid-Balance GFlowNet for Solving Vehicle Routing Problems
Venue: NeurIPS 2025
Accepted: yes
Keywords: GFlowNet

Vehicle Routing Problem
Abstract: Existing GFlowNet-based methods for vehicle routing problems (VRPs) typically employ Trajectory Balance (TB) to achieve global optimization but often neglect important aspects of local optimization. While Detailed Balance (DB) addresses local optimization more effectively, it alone falls short in solving VRPs, which inherently require holistic trajectory optimization. To address these limitations, we introduce the Hybrid-Balance GFlowNet (HBG) framework, which uniquely integrates TB and DB in a principled and adaptive manner by aligning their intrinsically complementary strengths. Additionally, we propose a specialized inference strategy for depot-centric scenarios like the Capacitated Vehicle Routing Problem (CVRP), leveraging the depot node's greater flexibility in selecting successors. Despite this specialization, HBG maintains broad applicability, extending effectively to problems without explicit depots, such as the Traveling Salesman Problem (TSP). We evaluate HBG by integrating it into two established GFlowNet-based solvers, i.e., AGFN and GFACS, and demonstrate consistent and significant improvements across both CVRP and TSP, underscoring the enhanced solution quality and generalization afforded by our approach.

Hybrid-Balance GFlowNet for Solving Vehicle Routing Problems

Abstract

Existing GFlowNet-based methods for vehicle routing problems (VRPs) typically employ Trajectory Balance (TB) to achieve global optimization but often neglect important aspects of local optimization. While Detailed Balance (DB) addresses local optimization more effectively, it alone falls short in solving VRPs, which inherently require holistic trajectory optimization. To address these limitations, we introduce the Hybrid-Balance GFlowNet (HBG) framework, which uniquely integrates TB and DB in a principled and adaptive manner by aligning their intrinsically complementary strengths. Additionally, we propose a specialized inference strategy for depot-centric scenarios like the Capacitated Vehicle Routing Problem (CVRP), leveraging the depot node's greater flexibility in selecting successors. Despite this specialization, HBG maintains broad applicability, extending effectively to problems without explicit depots, such as the Traveling Salesman Problem (TSP). We evaluate HBG by integrating it into two established GFlowNet-based solvers, i.e., AGFN and GFACS, and demonstrate consistent and significant improvements across both CVRP and TSP, underscoring the enhanced solution quality and generalization afforded by our approach.

1 Introduction

Vehicle Routing Problems (VRPs) are fundamental to real-world operations, including e-commerce logistics [54, 12, 39], urban delivery [56, 7, 21], supply chain management [11, 15, 8], and ridesharing systems [18, 30, 44]. Efficient VRP solutions directly affect cost reduction, service quality, and overall performance in transportation and supply chain networks. Over the past decades, numerous heuristic and meta-heuristic algorithms, such as the Lin-Kernighan-Helsgaun algorithm [19], ant colony optimization (ACO) [3], hybrid genetic search [46], tabu search [2], and simulated annealing [36], have been developed to address the combinatorial complexity of VRPs. However, these approaches often depend on handcrafted rules and problem-specific heuristics, which limit their adaptability and scalability across diverse VRP instances. More recently, reinforcement learning and deep learning methods have emerged as promising alternatives [38, 5, 52, 20]. Models such as POMO [25], NeuOpt [31], and DEITSP [47] show potential in reducing dependence on handcrafted components. Yet, these methods still struggle to consistently achieve desirable performance, often becoming trapped in local optima due to the limited exploration capacity.

To improve exploration, recent work has explored the use of Generative Flow Network (GFlownet) [4], which generate diverse and high-quality solutions through a probabilistic, generative process. Unlike traditional learning-based approaches that focus on optimizing a single or a few trajectories, GFlowNet aims to learn a distribution over the solution space, making them well-suited for combinatorial problems like VRPs. However, current GFlowNet-based methods for VRPs such as GFACS [22] and AGFN [55], rely exclusively on global optimization during training. Particularly, they both adopt the

Trajectory Balance (TB) objective [32], which effectively aligns with global metrics like minimizing total travel distance. However, this exclusive focus on global optimization can lead to the neglect of important local optimization signals. For instance, these methods may overlook reward dependencies between a current state and its successor, due to the lack of localized training objectives. As a result, they often struggle to capture fine-grained local structures in the solution space, limiting their ability to generate high-quality routes. On the other hand, Detailed Balance (DB) [4] offers a mechanism better suited for local optimization. Nevertheless, using DB alone is equally inadequate, as VRPs fundamentally require a global perspective to achieve optimal solutions. These limitations highlight the need for a broader approach that balances both local and global optimization. We propose that a Hybrid-Balance principle, combining TB and DB in a unified and extensible manner, can significantly enhance GFlowNet-based methods for VRPs.

Guided by this Hybrid-Balance principle, we introduce the Hybrid-Balance GFlowNet (HBG) framework for solving VRPs. First, HBG unifies DB, which promotes local optimization, with TB, which captures global optimization. To fully exploit their complementary strengths, we formulate a VRP-specific version of DB that effectively facilitates local optimization through localized objectives. We also design an adaptive integration mechanism that combines DB and TB in a way that respects their theoretical underpinnings, such as forward and backward transition probability, while leveraging their complementary benefits. Second, motivated by the insight that depot nodes in depot-centric VRPs, such as the Capacitated Vehicle Routing Problem (CVRP), have greater flexibility in selecting successor nodes, we propose a depot-guided inference strategy inspired by the Hybrid-Balance principle. Notably, even in depot-free scenarios like the Traveling Salesman Problem (TSP), our framework remains effective, as the Hybrid-Balance formulation is inherently general. Third, to demonstrate the broad applicability of HBG, we integrate it into two existing GFlowNet-based solvers, i.e., AGFN and GFACS, and observe consistent improvements in routing performance across both CVRP and TSP benchmarks. In summary, our main contributions are outlined as follows:

We propose the Hybrid-Balance GFlowNet (HBG) framework for solving VRPs, which, for the first time, introduces and formalizes the concept of DB within the VRP context. Meanwhile, it unifies the principles of TB and DB through a principled and coherent integration to process both local and global optimizations.

We design a depot-guided inference strategy to efficiently generate and explore high-quality trajectories, specifically tailored for problems involving a designated depot like the CVRP.

We incorporate the HBG framework into existing GFlowNet-based methods for solving VRPs, i.e., AGFN and GFACS, and evaluate it on both synthetic and real-world datasets. The results demonstrate that our method significantly improves the performance of GFlowNetbased solvers for CVRP and TSP.

2 Related Works

2.1 Learning-Based Solvers for Vehicle Routing Problems

Learning-based approaches for VRPs can generally be divided into two categories: constructionbased and improvement-based methods. Construction-based solvers generate complete solution trajectories in an end-to-end manner. A seminal example is the Attention Model (AM) [23], which first applied a Transformer architecture to solve VRPs. Building on AM, Policy Optimization with Multiple Optima (POMO) extends this approach by leveraging multiple optimal policies during training and inference to improve both solution quality and robustness. This line of work has since inspired a series of end-to-end construction-based methods [43, 26, 13, 47, 14] that further improve performance. Improvement-based methods, on the other hand, enhance initial solutions through iterative refinements. These methods often integrate neural networks into classical heuristic frameworks. Notable examples include NeuroLKH [49], DeepACO [51], and NeuOpt [31], which demonstrate strong performance through learning-augmented optimization strategies. To showcase the generality of our proposed framework, we apply it to enhance both a construction-based solver (AGFN) and an improvement-based solver (GFACS).

2.2 GFlowNet for Combinatorial Optimization Problems

GFlowNet has been applied across a wide range of structured generation and decision-making tasks. In molecular and drug discovery [57, 34, 24, 41, 16, 40], they are used to sample diverse, high-reward molecules from complex solution spaces. In causal structure learning [28, 9], GFlowNet facilitates exploration over multiple plausible directed acyclic graphs (DAGs), while in Bayesian inference [10, 42, 35], they serve as alternative samplers for discrete posteriors. Additional applications include symbolic reasoning [45, 27], robotics planning [29, 33], and solving maximum independent set (MIS) [53], where modeling solution diversity is essential. Recently, GFlowNet has also been applied to VRPs [55, 22], including TSP and CVRP. In this context, learning a distribution over feasible routes offers a flexible and effective alternative to deterministic solvers. Two representative models are AGFN and GFACS. AGFN incorporates adversarial training to improve trajectory construction in an end-to-end fashion, making it the first to apply GFlowNet to VRPs directly. In contrast, GFACS integrates GFlowNet with ant colony optimization, marking the first attempt to augment heuristic search with GFlowNet-based learning. In this paper, we further enhance both AGFN and GFACS for solving VRPs using our proposed Hybrid-Balance GFlowNet framework.

3 Hybrid-Balance GFlowNet

HBG

Loss

Current State S

t

Trajectory generation

Next State S

t+1

Trajectory

Local Search

GFACS

Energy

Reshaping

GFACS

TB Loss

Discriminator

AGFN

TB Loss

GFACS

AGFN

TB Training

v 1

v 2

…

Linear Layer

Z(S t+1

)

R(S t+1

)

Pf(S

t+1

)

Pb(S t+1

)

DB Loss

Pheromone Map

GFACS

Edge Probability

Distribution

AGFN

Generator

As illustrated in Fig.1, the Hybrid-Balance GFlowNet (HBG) begins with trajectory generation using either AGFN or GFACS, during which state transition information is recorded at each step. The generated next state is then treated as the current state in the following step, and this process repeats until a complete trajectory is constructed. The blue region in Fig.1 corresponds to the original components from AGFN and GFACS, responsible for the processing of the complete trajectory and the computation of the TB loss, which captures global optimization signals. However, relying solely on the TB loss can cause the model to overlook important relationships between individual states. To bridge this gap, our proposed HBG introduces additional components, highlighted in purple, where the DB loss is computed for each individual state transition to enhance local optimization. The final training objective is a combination of two loss, which together guide the optimization of the model.

To better illustrate the motivation for introducing DB, consider a long vehicle routing trajectory that is incrementally constructed as A → B → C → D → E → ... → U → V → W → X → Y → Z. Assume this complete route yields a high total cost (bad performance), primarily due to suboptimal decisions made in the early stages, such as traversing a high-cost edge from node C to node D. In contrast, the latter portion of the tour (e.g., from node U to Z) may follow a more cost-effective and well-structured pattern. Under Trajectory Balance (TB), the final reward is determined by the overall trajectory cost, and is proportionally assigned to all transitions. Consequently, even high-quality local transitions, such as W → X → Y, may receive weak or misleading training signals simply because they are embedded in a globally suboptimal trajectory. This would hinder the model's ability to learn and reinforce desirable local patterns. On the other hand, Detailed Balance (DB) operates at a

step-wise granularity, evaluating the expected outcomes of individual transitions. For instance, at node W, DB can assess whether transitioning to X leads to better outcomes compared to alternative choices like Z, regardless of earlier suboptimal steps. This localized and reward-sensitive feedback enables the model to more accurately learn local quality from global performance, and promotes stronger learning signals for valuable decisions even within imperfect trajectories.

This example illustrates a core limitation of Trajectory Balance (TB) in long-horizon combinatorial tasks like VRP: when the overall trajectory is suboptimal, TB lacks the ability to identify and preserve well-structured local segments within it. As a result, valuable local patterns may be overlooked or penalized. By incorporating Detailed Balance (DB) into the training objective, we address this limitation by providing fine-grained, step-level singal that helps isolate and reinforce high-quality local decisions, even when the global trajectory does not show good performance.

3.1 Modeling Basics

Problem Definition. For a CVRP instance, G denotes the input graph, which includes the coordinates and demands of customers, as well as the depot location. Formally, the instance is represented as a complete graph G = ( V , U ) , where V = { v 0 , v 1 , . . . , v n } denotes the set of nodes, with v 0 as the depot and the remaining nodes representing customers, and U is the set of edges. Each customer node v i ( i ≥ 1 ) is associated with a demand d i and a location in Euclidean space. Each edge ( v i , v j ) ∈ U has an associated cost c ij , typically defined as the Euclidean distance between v i and v j . The goal of CVRP is to determine a set of vehicle routes that start and end at the depot, such that each customer is visited exactly once, the total demand on each route does not exceed the vehicle's capacity, and the total routing cost is minimized.

State s : In a trajectory set T = { τ 1 , τ 2 , . . . , τ h } , the state s i denotes the sequence of nodes visited in trajectory τ i . At decision step t in τ i , the state is defined as s i t = { x i 0 , x i 1 , x i 2 , . . . , x i t } , where x i t is the most recently visited node, and x i 0 represents the depot which serves as both the starting and ending point of the route.

Action a : An action a i t transitions the system from state s i t to s i t +1 . Given s i t = { x i 0 , x i 1 , . . . , x i t } , the action selects the next node x i t +1 from the set of unvisited nodes, adhering to feasibility constraints such as vehicle capacity. Once all customers are visited, the route terminates in a final state, forming a complete trajectory τ i = { x i 0 , x i 1 , . . . , x i m } .

Reward R : The reward R ( τ i ) is determined by the quality of the generated trajectory τ i . We define two types of rewards: R ( τ i ) and R ( s i t ) . The former, R ( τ i ) , evaluates the entire trajectory, while the latter, R ( s i t ) , reflects local reward signals at individual state transitions. These are defined as: R ( τ i ) = ∑ m -1 k =0 d ( x i k , x i k +1 ) , R ( s i t ) = d ( x i t -1 , x i t ) , where d ( x i k , x i k +1 ) denotes the Euclidean distance between consecutive nodes.

Graph Neural Network (GNN). We integrate a GNN module [48] into the GFlowNet framework to more effectively capture the complex relational structures inherent in VRP instances. The detailed architecture and formulation are provided in Appendix A.1. Following the designs of AGFN and GFACS, we sparsify the fully connected graph into a k -nearest-neighbor graph G to improve scalability and reduce computational cost. The graph G is embedded into a high-dimensional feature space, encoding node coordinates and edge distances as node and edge features, respectively. The GNN, parameterized by θ , processes these features through multiple layers to produce rich representations. The resulting edge embeddings are passed through a multi-layer perceptron (MLP) to generate edge probability distribution η ( G ∗ , θ ) for decision making by GFlowNet, while the node embeddings Q = { q 1 , q 2 , . . . , q b } are retained for computing state flows.

3.2 Hybrid-Balance GFlowNet

3.2.1 Global Optimization via TB

In VRPs, the objective is to determine the shortest route while satisfying various operational constraints, which necessitates evaluating solutions from a global perspective. Both AGFN and GFACS adopt the Trajectory Balance (TB) objective to address this requirement, as it enables the GFlowNet to be trained over entire trajectories, naturally aligning with global optimization goals.

As illustrated in Fig. 1, AGFN generates an edge probability distribution η ( G ∗ , θ generator ) using GFlowNet, which is then used to sample the next node in the route. A discriminator, trained with false labels from GFlowNet-generated trajectories and true labels from near-optimal trajectories, evaluates the quality of sampled trajectory set T = { τ 1 , τ 2 , . . . , τ h } . It assigns a quality score to each trajectory, which is then combined with the raw trajectory length R ( τ ) to compute the final AGFN reward ˜ R ( τ ) . These rewards ˜ R ( τ ) , along with the source flow Z ( θ generator ) , forward probability P F ( τ ; θ generator ) , and backward probability P B ( τ ) obtained from the GFlowNet, are used to compute the AGFN TB loss ℓ AG TB , defined as:

For GFACS, the GFlowNet is used to generate a heuristic matrix η ( G ∗ , θ ) , which is subsequently transformed into a pheromone map to guide the ant colony optimization (ACO) in trajectory construction. Once the trajectories are generated, a local search is applied for refinement, followed by an energy reshaping step. The TB loss for GFACS, denoted ℓ GF TB , is then computed in a similar form to AGFN, as both approaches adopt the TB loss formulation to optimize their models.

3.2.2 Local-Global Optimization through Hybrid-Balance

While global optimization is essential for solving VRPs, local optimization is also important as it helps the model to capture fine-grained patterns, such as transitions between neighboring nodes. However, local information alone is insufficient for modeling global objective and constraints like total cost and capacity. To address this, we propose to unify both global and local objectives within a Hybrid-Balance GFlowNet framework. Specifically, we integrate the DB mechanism into the original TB framework of the GFACS and AGFN models to further enhance the modeling of local transitions, particularly the relationship between the current state s i t and the next state s i t +1 .

As shown in Fig. 1, the model records relevant information at each step, including the current state's reward R ( s i t ) and flow F ( s i t ; θ ) , the next state's reward R ( s i t +1 ) and flow F ( s i t +1 ; θ ) , as well as the forward and backward transition probability P f ( s i t +1 | s i t ; θ ) and P b ( s i t | s i t +1 ) . Once a trajectory is completed, we apply a forward-looking technique [37] to compute the DB loss ℓ DB between two successive states as:

Here, P f ( s i t +1 | s i t ; θ ) denotes the forward transition probability derived from the edge probability distribution η ( G ∗ , θ ) in AGFN or the pheromone map in GFACS. The relationship between the trajectory-level forward probability P F ( τ i ; θ ) used in TB loss and the step-wise forward probability P f ( s i t +1 | s i t ; θ ) used in DB loss is given by:

To ensure consistency with the trajectory-level backward probability P B ( τ i ) used in TB loss, we design the step-wise backward probability P b ( s i t | s i t +1 ) to reflect the structure of sub-trajectories within τ i . Specifically, we assume that each complete trajectory τ i consists of a multi-node subtrajectories and j single-node sub-trajectories, and parameter P b is accordingly determined by the varied transition structures.

Definition 1 (Trajectory Composition and Ordering Count) . We define A a as the set of a multinode trajectories, and J j as the set of j single-node trajectories. Together, these sequences are combined to form a complete trajectory τ i . Let B ( A a , J j ) denote the number of distinct orderings of sub-trajectories in A a and J j that result in the same complete trajectory τ i .

We next present the following statement, which describes the recurrence relation for B ( A a , J j ) .

Statement 1 (Trajectory Orders' Count Recurrence) . The number of distinct trajectories composed of a multi-node trajectories and j single-node trajectories arranged in different orders, denoted by B ( A a , J j ) , satisfies the following recurrence relation:

This recurrence arises from the backward destruction of CVRP trajectories, where we consider how B ( A a , J j ) reached its predecessors. Suppose the current state corresponds to B ( A a , J j ) , where there are a remaining multi-node trajectories and j remaining single-node trajectories to be disconnected. There are two possible types of backward transitions from this state to reach its predecessor B ( A a -1 , J j ) or B ( A a , J j -1 ) :

(1) Multi-node trajectory: If a multi-node trajectory is selected for backward destruction from the depot, either of its two nodes can serve as the immediate predecessor to the depot. Therefore, each of the a multi-node trajectories contributes two valid backward transitions, resulting in a total contribution of 2 a · B ( A a -1 , J j ) , where the recursion proceeds with a -1 remaining multi-node trajectories and j unchanged single-node trajectories.

(2) Single-node trajectory: If a single-node trajectory is chosen, it contains only one node, which uniquely determines the depot's predecessor. Thus, each of the j single-node trajectories contributes one backward transition, resulting in j · B ( A a , J j -1 ) , where the recursion continues with a multi-node trajectories and j -1 single-node trajectories.

We combine both types of transitions to derive the recurrence relation as presented in Eq. 4. Subsequently, we deduce the closed-form expression of B ( A a , J j ) from Eq. 4. The proof is provided in Appendix Sec. A.2, and the corresponding formulation is presented below:

Physically, the term ( a + j )! accounts for all possible orderings of the a + j sub-trajectories, where each of them is treated as an atomic step in the destruction process. Each multi-node trajectory has 2 possible directions for destruction, contributing an additional 2 a multiplicative factor. In contrast, single-node trajectories allow only one valid direction. Therefore, the total number of reward-equivalent permutations is the product of these two factors.

Statement 2 (TB Backward Probability) . We denote P B ( τ i ) as the backward policy probability of a complete trajectory τ i in the GFlowNet framework under TB, formulated as:

where the denominator reflects the total number of distinguishable trajectory permutations given a multi-node and j single-node trajectories to achieve complete trajectory τ i .

Statement 3 (DB Backward Probability) . We denote P b ( s i t | s i t +1 ) as the probability of a single backward transition from state s i t +1 to its predecessor s i t , and under the DB formulation, the backward probability is defined conditionally:

This probability formulation originates from Eq. 4, which defines the total number of sub-trajectory backward destruction orderings. Physically, each multi-node trajectory offers two possible predecessor nodes for backward disconnection from the depot, thereby contributing the 2 a term, while each single-node trajectory provides one such option, contributing the j term. The resulting probability 1 2 a + j reflects a uniform selection over all valid backward transitions at the current decision step. In contrast, for all other nodes in the trajectory, only a single predecessor is feasible, and thus the backward transition becomes fully deterministic with probability 1 .

Meanwhile, F ( s i t ; θ ) in Eq. 2 represents the flow of current state, and is derived from the node embedding q at state s i t , which is calculated as follow:

where W 1 , W 2 , b 1 and b 2 are learnable parameters and ReLU [6] is the activation function. To handle the local objective associated with state transitions, we define the reward of the predecessor state s i t as zero. Consequently, the energy term ˜ E ( s i t ) in Eq. 2 is also set to zero. The successor state s i t +1 , in contrast, receives a non-zero transition reward. Accordingly, the energy term ˜ E ( s i t +1 ) represents the local reward signal, and its negative is defined as follow:

As training progresses, the quality of each trajectory steadily improves, resulting in smaller values of R ( s i t ) as the generated routes become shorter. In Eq. 9, we compute the energy ˜ E ( s i t ) for state s i t by subtracting the average reward of other trajectories at the same decision step. This formulation effectively captures the relative advantage of a given state compared to its peers, encouraging the model to assign higher energy to better-performing states. As the variance across rewards decreases during training, the energy values naturally increase.

Then, the DB loss of the completed trajectory τ i = { x i 0 , x i 1 , x i 2 , . . . , x i m } can be calculated as:

where ℓ DB ( s i t , s i t +1 ; θ ) is derived from Eq. 2. The overall loss for the Hybrid-Balance GFlowNet, denoted by ℓ HB ( T ; θ ) , is computed by aggregating both the TB loss ℓ TB ( τ ; θ ) and the DB loss ℓ DB ( τ ; θ ) over all trajectories:

This unified objective enables the model to simultaneously capture global trajectory-level structure and fine-grained local transitions, leading to more effective and robust optimization in VRPs.

3.2.3 Depot-Guided Inference

Current Node is Customer

Discrete Node

Current Discrete Node

Next Discrete Node Candidate

Other Node

Current Destruction Trajectory

Future Destruction Trajectory

The design of Hybrid-Balance GFlowNet's backward policy reveals a key insight: as illustrated in Fig. 2, only the depot node retains flexibility in choosing among multiple predecessor candidates during trajectory destruction. This flexibility stems from the construction of sub-trajectories, each of which begins and ends at the depot. In contrast, for all customer nodes, the backward transition path is uniquely defined by the trajectory structure, i.e., once a customer node is reached, its predecessor is deterministically identified. This determinism also holds during forward trajectory construction. To leverage this structural characteristic, we propose a depot-guided inference mechanism defined as:

where x ∼ P f ( s t +1 | s t ; θ ) denotes sampling from the forward transition probability, and x ∗ = arg max P f ( s t +1 | s t ; θ ) corresponds to greedy selection. Here, P f ( s t +1 | s t ; θ ) is derived from

the edge probability distribution η ( G ∗ , θ ) in AGFN or the pheromone map in GFACS. Under this strategy, exploration through sampling is applied only at the depot, while customer nodes follow a deterministic, greedy policy.

It is important to note that depot-guided inference is specifically designed for problems featuring a designated depot node, such as the CVRP. For problems lacking a depot or node-role differentiation, such as the TSP, we retain their original inference procedures, including hybrid decoding strategy [55] for AGFN and the ant clony search [22] for GFACS.

4 Experiment

We conduct experiments to validate the effectiveness of the Hybrid-Balance GFlowNet (HBG) in enhancing two representative GFlowNet-based solvers, i.e., AGFN and GFACS, on CVRP. We first present comparison results, followed by ablation studies to analyze the contribution of individual components. Lastly, we extend the evaluation to other vehicle routing problem.

Dataset: We adopt synthetic CVRP datasets following standard settings used in prior work [22, 25, 55, 49]. Each instance features a single depot and multiple customers served by a vehicle with fixed capacity C . The depot and customer coordinates are sampled uniformly from the unit square [0 , 1] 2 , and customer demands follow a uniform distribution U [ a, b ] with a = 1 and b = 9 . The vehicle capacity is fixed at C = 50 across all problem sizes: 100, 200, 500, and 1,000 nodes. For testing, we generate 128 synthetic instances for each of the 200-, 500-, and 1,000-node settings, aligned with evaluation rules established in AGFN and GFACS. The code is available at https: //github.com/ZHANG-NI/HBG

Hyperparameters: We adopt the same model configurations and training settings as AGFN and GFACS, including network architecture, batch size, learning rate, optimizer, and other hyperparameters. Training is conducted using sampling-based decoding with N = 20 routes per instance. During inference, AGFN uses depot-guided inference, and GFACS applies an ant colony search with depot-guided node selection. All models are trained on 100-node instances. The experiments are conducted on a server equipped with an NVIDIA A100 GPU and an Intel Xeon 6342 CPU.

4.1 Performance on Synthetic CVRP Instances

We compare HBG-enhanced models, i.e., HBG-AGFN and HBG-GFACS, with their original TBbased counterparts, AGFN [55] and GFACS [22]. AGFN constructs routes in an end-to-end manner, while GFACS searches for solutions by combining GFlowNet with ant colony optimization. We also include classical heuristics (LKH [19], ACO [3]) and learning-based baselines (POMO [25], GANCO [50], NeuOpt [31]) for comparison. All methods are trained on 100-node instances and evaluated on CVRP200, CVRP500, and CVRP1000 datasets, following AGFN and GFACS evaluation protocols. Additional experiments on the public benchmark CVRPLib are reported in Appendix B.1.

Table 1 shows that HBG consistently improves performance across all problem sizes for AGFN, GFACS, and GFACS with local search. The performance gains are significant, with gap reductions of up to 16.23%, 55.46%, and 8.33%, respectively. Improvements become more pronounced as instance size increases, indicating strong scalability. Inference incurs only minor overhead (0.01-0.04

(a) AGFN

(b) GFACS

seconds) due to temporary loading of flow parameters, which does not impact overall runtime or scalability. Compared to other heuristic and learning-based methods, HBG-AGFN and HBG-GFACS achieve competitive or superior solution quality across all scales. On CVRP200, both methods outperform ACO and NeuOpt. On CVRP500 and CVRP1000, they continue to generalize effectively, outperforming ACO, POMO, GANCO, and NeuOpt. These results highlight the robustness, efficiency, and strong generalization capabilities of the proposed HBG framework.

4.2 Ablation Study

Comparison of Component Contributions. We evaluate the contribution of each component in the HBG framework for both AGFN and GFACS. First, we incorporate the Hybrid-Balance (HB) module into the original models. Then, we add the depot-guided inference mechanism on top of the HB-enhanced variants. As shown in Table 2, each component contributes significantly to performance. Incorporating the HB module alone reduces the optimality gap by up to 15.07% in AGFN and 16.58% in GFACS. Adding depot-guided inference provides further gains, especially for larger instances. These results confirm that the HB module offers consistent improvements and depot-guided inference delivers additional benefits in depot-centric tasks.

Comparison of Balance Strategies. To further validate the effectiveness of Hybrid Balance (HB), we conduct a comparison against Trajectory Balance (TB) and Detailed Balance (DB) under identical training settings on 100-node instances, evaluated on CVRP200, CVRP500, and CVRP1000. As shown in Table 3, the HB module consistently outperforms both TB and DB across all instance sizes for both AGFN and GFACS. Notably, HB achieves up to a 15.07% improvement over TB in AGFN and up to 16.58% in GFACS. These results highlight the superior effectiveness of Hybrid Balance as a unifying optimization strategy.

Depot-Guided Inference Variants. We assess four variants of the depot-guided inference strategy by applying either sampling or greedy decoding at the depot and customer nodes. Tests are conducted using both AGFN and GFACS on CVRP200, CVRP500, and CVRP1000. As shown in Table 4, the combination of sampling at the depot and greedy decoding at customers yields the best performance. This setting consistently outperforms all other variants, including depot greedy + customer sampling, depot greedy + customer greedy, and depot sampling + customer sampling. These results validate the effectness of our depot-guided inference mechanism.

4.3 Generalization to Other Vehicle Routing Problem

We further evaluate our framework on the Traveling Salesman Problem (TSP), a key VRP variant. Baselines include GFlowNet-based solvers (AGFN [55], GFACS [22]), classical heuristics (LKH [19], ACO [3]), and learning-based models (POMO [25], GANCO [50], NeuOpt [31]). All models are trained on 100-node instances and evaluated on 200-, 500-, and 1,000-node settings. Since TSP lacks a depot node, depot-guided inference is not used. Table 5 shows that HBG-AGFN consistently outperforms AGFN, reducing the gap by up to 17.64%. HBG-GFACS also achieves

(b) GFACS

notable improvements, with the gap on 200-node instances reduced from 22.79% to 19.40%. With local search, HBG-GFACS achieves further improvements, with the best gap reduction reaching 6.31%. Compared to classical heuristics and learning-based methods such as ACO, POMO, NeuOpt, and GANCO, both HBG-AGFN and HBG-GFACS achieve competitive results on TSP tasks. These results confirm the generalizability and strong performance of HBG on TSP tasks.

5 Conclusion

In this paper, we introduced the Hybrid-Balance GFlowNet (HBG) framework to enhance the performance of GFlowNet-based solvers for vehicle routing problems. HBG unifies Trajectory Balance and Detailed Balance in a principled and adaptive manner to jointly optimize local and global objectives. We also proposed a depot-guided inference strategy aligned with the Hybrid-Balance principle, specifically tailored for depot-centric problems. Extensive experiments on both CVRP and TSP benchmarks demonstrate that HBG significantly improves the performance of two representative GFlowNet-based solvers, i.e., AGFN and GFACS, showcasing improved solution quality, scalability, and generalization. A current limitation of HBG is its reliance on existing GFlowNet-based models, as its performance depends in part on the underlying solver, which might be inferior to others. In future work, we plan to integrate it with alternative stronger generative policies and solvers.

Acknowledgments and Disclosure of Funding

This research is supported by the National Research Foundation, Singapore under its AI Singapore Programme (AISG Award No: AISG3-RP-2022-031, AISG3-RP-2025-036-USNSF), and the Lee Kong Chian Fellowship awarded to CAO Zhiguang by Singapore Management University.

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A Methodology Details

A.1 Graph Neural Network

Our GNN architecture follows the same design as those used in AGFN and GFACS to ensure a fair comparison, which uses a custom GNN architecture designed specifically for the VRP task, and employs a custom message-passing GNN that jointly updates node and edge representations over multiple layers. At each layer l , node embedding h l i of the i -th node and edge embedding e l ij between the i -th and j -th nodes are updated via the following formulations:

where W l 1 , W l 2 , W l 3 , W l 4 , W l 5 are learnable parameters, A ( · ) denotes the aggregation function (mean pooling in our case), σ ( · ) is the sigmoid activation that modulates attention over neighbors, and ACT denotes the SiLU activation. Batch normalization (BN) is applied at each step for stability. The number of layers is set to 12 for HBG-GFACS and 16 for HBG-AGFN, with hidden dimensions of 32 and 64, respectively.

A.2 Proof Process of Hybrid-Balance

To solve Eq. 4, We begin by establishing the boundary conditions. When either a = 0 or j = 0 , the recurrence simplifies accordingly. For instance, when a = 0 , we obtain:

When a = 0 and j = 0 , there are no sub-trajectories in τ i , and τ i contains only the depot node. Therefore, the base case becomes B ( A 0 , J 0 ) = 1 , and it is equivalent to:

Using Eq. 15 recursively and applying the base case B ( A 0 , J 0 ) = 1 , we derive:

Similarly, when j = 0 , we can deduce that:

We now normalize Eq. 4 by dividing both sides by 2 a · a ! · j ! , resulting in:

Based on this, we define a normalized function:

Substituting Eq. 20 into Eq. 19, we obtain the recurrence:

We analyze the recurrence in Eq. 21 under the boundary conditions, and derive results from Eqs. 17, 18, and 20:

These boundary conditions are equivalent to:

Moreover, motivated by Pascal's rule [17], which states:

and by setting n = a + j , k = a , we obtain:

which is equivalent to:

This is similar with the function of c ( A a , J j ) in Eq. 21. Therefore, the closed-form expression of c ( A a , J j ) is:

By extending Eqs. 16, 22, 23, and 24, the generalized closed-form solution for all a, j ≥ 0 is given as:

Finally, substituting Eq. 25 into Eq. 20, we obtain the closed-form expression for the total number of distinct sub-trajectory orderings resulting in the same complete trajectory:

B More Experiments

B.1 Test on Real world dataset

To evaluate our model's performance on real-world data, we conduct experiments on the CVRPLIBXXL [1], which is designed to test model's performance on large-scale real-world instances. As shown in Table 6, HBG-AGFN achieves up to a 91.51% reduction in gap compared to AGFN, while HBG-GFACS achieves up to 98.52% improvement over GFACS. These results demonstrate that our framework significantly enhances the performance of GFlowNet-based solvers on real-world datasets.

(a) Sparsity Parameter

(b) Learning Rate

(c) Optimizer Type

B.2 Hyperparameter Sensitivity Analysis on CVRP

The introduction of a new loss component ( L DB ) could potentially alter the optimization landscape, and that the original hyperparameter settings used in AGFN and GFACS may not be optimal for the proposed HBG variants. We further conduct additional experiments in which we re-tuned key hyperparameters for the HBG models, including the sparsity parameter, learning rate, and optimizer settings. The updated results (Table 7a-7c) will be incorporated into the Appendix of the revised version. HBG-GFACS_2 refers to the HBG-GFACS model evaluated with a sparsity parameter k = | V | / 2 ; the interpretation is analogous for the other entries. The term origin denotes the original hyperparameter configuration used in our main experiments. HBG-GFACS_5 × 10 -3 refers to the HBG-GFACS model evaluated with a learning rate of 5 × 10 -3 ; the interpretation is analogous for the other entries. HBG-GFACS_SGD refers to the HBG-GFACS model evaluated using the SGD optimizer; the interpretation is analogous for the other entries.

As the sparsity parameter results shown in Table 7a, the optimal sparsity setting yields the best performance for both HBG-GFACS and HBG-AGFN. Regarding the learning rate comparisons in Table 7b, for HBG-GFACS, the original value of 5 × 10 -4 performs best without local search, while a value of 1 × 10 -3 achieves the best results when local search is enabled. For HBG-AGFN, a learning rate of 1 × 10 -4 shows superior performance on the 200- and 500-node instances, whereas the original value performs best on the 1000-node instances. As for the optimizer comparison in Table 7c, the original AdamW setting provides the best performance in most cases. An exception is observed on the 500- and 1000-node instance with local search, where Adam slightly outperforms AdamW for HBG-GFACS.

B.3 Weight λ of DB and TB

We have conducted additional experiments on the weighted combination (i.e., L HB = L TB + λ L DB ) to assess the sensitivity of our method to different values of λ , which are gathered in Tables 8-11. The results include both adaptive weights (i.e., λ : 0.5 → 0, 1 → 0, 2 → 0 and so on) and fixed weights (i.e., 0.5 → 0.5, 1 → 1, 2 → 2), where 0.5 → 0 denotes changing the TB:DB loss weight ratio from 1:0.5 to 1:0 during training, with TB weight fixed at 1. The interpretation for other entries follows analogously. The term origin denotes the original hyperparameter setting used in our main experiments. We find that, for HBG-AGFN, a fixed 1:1 weight between TB and DB consistently yields the best performance. Similarly, for HBG-GFACS, this ratio offers the most favorable trade-off between performance and stability with and without local search.

B.4 Statistical Significance of Experiment

2000

4000

6000

8000

10000

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8

10

12

14

16

18

20

Obj.

HBG-AGFN

200

400

600

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7.9

8.0

8.1

8.2

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HBG-GFACS

As shown in Fig. 3, both HBG-AGFN and HBG-GFACS exhibit a clear and steady decline in objective values throughout the training process, indicating stable convergence. The narrow shaded areas-representing standard deviation across five random seeds-suggest low variance among runs. These results collectively highlight the effectiveness of the training process and the statistical reliability of the observed performance gains.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper discusses 2 different methods for gflownet that are limited, Trajectory Balance (TB) objective effectively aligns with global metrics of VRP and TSP like minimizing total travel distance. However, this exclusive focus on global optimization can lead to the neglect of important local optimization signals.  On the other hand, Detailed Balance (DB) [4] offers a mechanism better suited for local optimization. Nevertheless, using DB alone is equally inadequate, as VRPs fundamentally require a global perspective to achieve optimal solutions. For this reason they propose Hybrid-Balance principle, combining TB and DB in a unified and extensible manner, can significantly enhance GFlowNet-based methods for VRPs.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n1. Conceptually Sound and Well-Motivated: The central premise of the paper—that VRPs benefit from both global and local optimization—is highly intuitive and well-articulated. The authors clearly explain the complementary strengths and individual weaknesses of Trajectory Balance (global) and Detailed Balance (local), providing a strong motivation for their hybrid approach. \n\n2. Novel Theoretical Contribution: The paper's most significant strength lies in its formalization of Detailed Balance for VRPs. The derivation of the backward transition probability,  is non-trivial. The authors provide a combinatorial argument (Statements 1, 2, and 3) to derive a closed-form expression for the number of trajectory orderings, which then informs the backward probability at the depot.\n\n3. Comprehensive and Rigorous Evaluation: The experimental setup is thorough. By integrating HBG into two distinct GFlowNet-based solvers—an end-to-end construction method (AGFN) and a hybrid heuristic search method (GFACS)—the authors demonstrate the general applicability of their framework. The evaluation across two different problems (CVRP and TSP) and multiple scales (up to 1000 nodes) further strengthens the claims of robustness and scalability.\n\nWeaknesses\n\nMy main concerns with this paper are the claimed \"adaptiveness\" of the hybrid balance, the practical significance of some of the performance gains, and the potential overstatement of novelty regarding the inference strategy.\n\n1. \"Hybrid-Balance\" is a Simple, Static Summation: The paper claims to integrate TB and DB in a \"principled and adaptive manner\". However, the final loss function in Equation 11 is a simple, unweighted sum: LossHB​=LossTB​+LossDB​. This is the most straightforward, non-adaptive way to combine two losses. There is no adaptive mechanism described that would, for example, change the weighting between LossTB​ and LossDB​ during training or based on the problem instance. This makes the term \"adaptive\" misleading. A simple summation might not be optimal, and this lack of exploration into the balance mechanism is a significant weakness.\n\n2. Marginal Performance Gains in Some Key Scenarios: While the improvements are consistent, their magnitude is questionable in some cases. For the end-to-end AGFN model on CVRP1000, the objective value improves from 133.97 to 131.78, a gain of only ~1.6%.\n\n3. Fairness of Hyperparameter Tuning: The authors state they \"adopt the same model configurations and training settings as AGFN and GFACS\". However, the introduction of a new loss component (lDB​) fundamentally changes the optimization landscape. It is highly likely that the optimal hyperparameters (e.g., learning rate, optimizer settings) for the HBG models would be different from the original models. Without re-tuning, the comparison might unfairly handicap the proposed HBG models, or conversely, the original models might have been better tuned for their specific loss. The paper provides no information on whether such re-tuning was performed.\n\nQUESTIONS:\n1. The term \"adaptive\" is used to describe the integration of TB and DB, yet Equation 11 shows a fixed, unweighted sum. Could you clarify what aspect of the integration is adaptive? Have you experimented with a weighted sum, lTB​+λ⋅lDB​, and if so, how sensitive is the performance to the hyperparameter λ?\n\n2. The energy in Equation 9 is based on the reward relative to the batch average. What is the intuition behind this choice versus a more direct energy function\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the Hybrid-Balance GFlowNet (HBG) framework aimed at improving GFlowNet-based solvers for vehicle routing problems (VRPs). HBG combines trajectory balance (TB) and detailed balance (DB) objectives in a principled, adaptive manner to unify global and local optimization perspectives. The authors design a depot-guided inference scheme tailored for VRPs with a depot structure (e.g., CVRP), and rigorously evaluate the framework by integrating HBG into two state-of-the-art GFlowNet-based solvers: AGFN and GFACS. Experimental results demonstrate significant and consistent improvements in solution quality and generalizability.\n\nSTRENGTHS AND WEAKNESSES:\n## Strength\n- The integration of TB and DB provides a unified approach bridging the gap between local and global optimization for GFlowNets in combinatorial domains. The formalization of DB within VRP, particularly the explicit recurrence  and probability derivation, is a notable theoretical advancement.\n- The main development, including Algorithmic design, mathematical derivations, and the final composite loss function, is clearly presented.\n- Empirical improvements are substantial and consistent, with ablation studies confirming the contribution of each module. The improvements persist for TSP, underlining generalizability. The evaluation of depot-guided inference variants is carefully done and matches the method’s claimed benefits.\n\n## Weakness\n- The necessity of considering  local optimization signals is not well-motivated. In particular, why we also need to pursue local optimization given a global optimization objective? Will the local optimization signals bias the pursuit of global optimum? What does \"fine-grained local structure\" refer to?\n- It is noteworthy that the depot-guided inference strategy only applies to depot-centric problems like CVRP. For problems without clear depot structures (like TSP), the method falls back to original approaches.\n- The performance could be constrained by the quality of the base GFlowNet solvers.\n\nQUESTIONS:\nsee my comments in the Strengths And Weaknesses section\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a novel learning-based approach for Solving Vehicle Routing problems and builds on GFlowNet based methods for finding a distribution over potential solutions. While existing approaches usually optimize towards a global trajectory constraint (Trajectory Balance), the paper extends this formulation with a local inter-state constraint called Detailed Balance. The authors claim that solely TB often shows limiting capabilities in local optimization and DB on the other hand fails on global optimization, which is an integral part of Vehicle Routing Problems. The authors propose a method that includes both optimization objectives and show improved results on standard benchmarks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n- The overall presentation of the methodology is technical sound and the authors provide the many details of the proposed methodology.\n- The results in Table 1 indicate an improvement over the previous approaches (AGFN, GFACS).\n- The authors provide an insightful ablation study for the balancing strategies and the component contributions.\n\nWeaknesses\n\n- The Detailed Balance constraint needs to be better motivated. It is unclear why there is a problem with TB and  the local optimization objective should help for this global optimization problem. Usually an insightful example from the test data can help with that and further strengthen the story of the paper since the DB is the main contribution of the paper.\n- The authors decided to have the Related Works section in the Appendix. As a reader who is not entirely aware of this branch of research, needs to directly jump to the appendix before reading the full paper. I’d suggest shortening the introduction and the method section to have the related work section in the main paper.\n- Some in the methodology section of the paper are hard to understand, e.g. In equation 2, there is an energy term which is discussed two pages later for the first time. According to equation 9 the energy can be negative if R is larger than the average R. Is this an intended behaviour?\n- Most of the approach is adapted from previous works (Trajectory Generation, TB training) and the contribution is mainly to extend to the objective with the local DB constraint for this specific vehicle routing task. I believe that this approach could also serve for other combinatorial optimization problems.\n\nQUESTIONS:\nPlease comment on the weaknesses.\n\nLIMITATIONS:\nPlease discuss limitation in more detail in the Appendix.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces Hybrid-Balance GFlowNet (HBG), a framework that unifies Trajectory Balance (TB) and Detailed Balance (DB) objectives for solving Vehicle Routing Problems. The authors argue that existing GFlowNet-based VRP solvers like AGFN and GFACS rely exclusively on TB, which captures global optimization but neglects local optimization signals. Their key insight is that DB can address local optimization while TB handles global objectives. The framework includes a depot-guided inference strategy that exploits the structural property that depot nodes have greater flexibility in successor selection compared to customer nodes. Experimental results on CVRP and TSP show consistent improvements when HBG is integrated into existing solvers.\n\nSTRENGTHS AND WEAKNESSES:\n__Strengths:__\n\nThe observation that existing GFlowNet VRP solvers focus exclusively on global optimization through TB while ignoring local patterns is valid and represents a genuine limitation. \n\nThe observation that depot nodes have multiple valid predecessor options during backward trajectory destruction, while customer nodes have deterministic predecessors is well-exploited.\n\nThe experimental evaluation covered multiple problem instances and baseline comparisons. The consistent improvements across both AGFN and GFACS demonstrate the generality of the approach. \n\n \n\n__Weaknesses:__\n\nThe paper essentially combines two existing GFlowNet objectives (TB and DB). While the combination is sensible, the theoretical contribution is limited.\n\nThe relationship between the step-wise and trajectory-level probabilities could be better explained, particularly how the DB formulation maintains consistency with the overall GFlowNet framework.\n\nQUESTIONS:\nThe paper's treatment of the Graph Neural Network component lacks crucial implementation details and theoretical motivation. While Section 2.1 mentions incorporating a GNN module to 'effectively model complex relational structures of VRP instances,' several important aspects remain unclear:\n\n1. The paper cites a general GNN survey [45] but does not specify which GNN variant is employed (e.g., GCN, GraphSAGE, GAT, or a custom architecture). This omission makes reproduction difficult.\n\n2. How exactly does information propagate between nodes? What are the specific update functions for node and edge representations? The paper mentions processing features 'through multiple layers' but provides no details about the aggregation and update schemes.\n\n3.  While k-nearest-neighbor sparsification is mentioned for scalability, what value of k is used? How is this parameter selected, and how does it affect performance across different problem sizes?\n\n4. The experiments section lacks any GNN-specific hyperparameters (number of layers, hidden dimensions, activation functions, normalization schemes). These details are essential for reproducibility and understanding the method's sensitivity to architectural choices.\n\nAlso, the combination of TB and DB losses through simple summation (Equation 11), have the authors considered adaptive weights?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 1, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-18"}