FiLMMeD: Feature-wise Linear Modulation for Cross-Problem Multi-Depot Vehicle Routing Arthur Corrêaa,∗ , Paulo Nascimentoa and Samuel Moniza
arXiv:2604.28102v1 [cs.LG] 30 Apr 2026
a University of Coimbra, CEMMPRE, ARISE, Department of Mechanical Engineering, Coimbra, 3030-788, Portugal
ARTICLE INFO
ABSTRACT
Keywords: Vehicle routing Multi-task learning Cross-problem generalization Curriculum learning Preference optimization
Solving practical multi-depot vehicle routing problems (MDVRP) is a challenging optimization task central to modern logistics, increasingly driven by e-commerce. To address the MDVRP’s computational complexity, neural-based combinatorial optimization methods offer a promising scalable alternative to traditional approaches. However, neural-based methods typically rely on rigid architectures and input encodings tailored to specific problem formulations. In real-world settings, heterogeneous constraints create multiple MDVRP variants, limiting the applicability of such models. While multi-task learning (MTL) has begun to accelerate the development of unified neural-based solvers, prior works focus almost exclusively on single-depot VRPs, leaving the MDVRP unaddressed. To bridge this gap, we propose Feature-wise Linear Modulation for Cross-Problem Multi-Depot Vehicle Routing (FiLMMeD), a novel unified neural-based model for 24 different MDVRP variants. We introduce three main contributions: (1) to improve the model’s generalization, we augment the standard Transformer encoder with Feature-wise Linear Modulation (FiLM), which dynamically conditions learned internal representations based on the active set of constraints; (2) we provide an initial demonstration of Preference Optimization in the MTL setting, establishing it as a superior alternative to Reinforcement Learning for future MTL works; (3) to mitigate the generalization gap caused by the introduction of multi-depot constraints, we introduce a targeted curriculum learning strategy that progressively exposes the model to increasingly more complex constraint interactions. Extensive experiments on 24 MDVRP variants (including 8 novel formulations) and 16 singledepot VRPs confirm the effectiveness of FiLMMeD, which consistently outperforms state-of-the-art baselines. Our code is available at: https://github.com/AJ-Correa/FiLMMeD/tree/main
1. Introduction The vehicle routing problem (VRP) belongs to a class of canonical combinatorial optimization (CO) problems that hold wide importance in both the operations research and computer science communities. VRPs arise in numerous real-world applications, such as logistics and drone delivery (Cattaruzza et al., 2017; Li et al., 2019; Wang and Sheu, 2019). On one hand, due to the inherent NP-hard nature of the VRP, obtaining optimal solutions with exact algorithms is typically intractable for instances with more than a couple hundred nodes (Pessoa et al., 2020). On the other hand, heuristic algorithms offer a practical compromise, achieving (near-)optimal solutions within manageable runtimes, but their design and tuning often depend on specialized expert knowledge. Recently, neural-based approaches have been gaining traction as an alternative approach to solve VRPs (Bengio et al., 2021; Bogyrbayeva et al., 2024). These methods leverage deep learning to approximate powerful policies, capable of finding high-quality solutions with minimal computational overhead and domain expertise. However, like conventional approaches, most existing neuralbased methods are tailored and trained for specific types of VRP variants, which limits their applicability across diverse scenarios (Kool et al., 2019; Kwon et al., 2020; Zhou et al., 2023). ∗ Corresponding author
[email protected] (A. Corrêa); [email protected] (P. Nascimento); [email protected] (S. Moniz) ORCID (s): 0009-0000-4949-4791 (A. Corrêa); 0000-0003-4139-1100 (P. Nascimento); 0000-0002-7813-4514 (S. Moniz)
A. Corrêa et al.
1.1. Motivation Among existing VRP variants, the multi-depot VRP (MDVRP) is of particular practical relevance. It extends the classic VRP by considering multiple depots from where vehicles are dispatched. This setting has become increasingly important with the rise of e-commerce and growing customer expectations for faster deliveries, prompting companies to operate multiple depots (Lyu et al., 2023). Despite that, the MDVRP has been largely neglected by the neuralbased CO literature, where most efforts have focused on the classic single-depot VRP. Furthermore, considering the sheer amount of existing VRP variants (Elatar et al., 2023), each defined by distinct sets of operational constraints, designing and training separate models for each one is impractical. Crucially, in real-world logistics, these constraints are rarely static. A delivery fleet might face different daily requirements, from varying depot availability and time windows to backhaul requests and route length limits (Pillac et al., 2013) — depending on customer needs, seasonality factors or regulatory changes. Notwithstanding this fact, traditional solution methods are typically designed for fixed problem structures and constraints (Braekers et al., 2016), lacking the flexibility to adapt to different operating settings, often requiring a separate method for each scenario. Consequently, there is an existing need for a single unified routing model capable of solving diverse constraint combinations without being retrained each time. To address this, the literature has recently started to move towards the development of unified cross-problem Page 1 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
models through the application of multi-task learning (MTL) (Liu et al., 2024; Zhou et al., 2024a; Berto et al., 2025). MTL is a machine learning paradigm that allows a single model to learn from multiple related tasks simultaneously, leveraging shared representations to capture common structural patterns across them (Zhang and Yang, 2022). In the context of VRPs, this allows a model to generalize across different problem variants without having to be retrained for each formulation. In spite of recent efforts, prior MTL studies have not explicitly targeted MDVRP variants. While RouteFinder (Berto et al., 2025) and CaDA (Li et al., 2025) have addressed MDVRPs to some extent, they operate in a zero- or few-shot learning manner, leading to substantial performance gaps relative to traditional non-neural baselines. In addition, existing MTL models do not structurally condition learned representations based on the active constraints of each VRP variant. Although CaDA introduced constraint prompts, it relies on static feature concatenation that treats constraints as high-level context rather than directly modulating node embeddings. As a result, existing models still fail to distinguish the structural requirements of each variant, inducing gradient conflicts that limit their generalization across diverse constraint sets. Lastly, prior MTL works have relied on relatively simple training strategies. For example, MTPOMO (Liu et al., 2024) and MVMoE (Zhou et al., 2024a) train almost exclusively on variants with a single active constraint. Conversely, RouteFinder (Berto et al., 2025) samples all available variants simultaneously, regardless of the number of active constraints. While these strategies work well for single-depot VRPs, our experiments have shown that they are insufficient to achieve good generalization on MDVRP variants.
1.2. Our contributions To mitigate the aforementioned challenges, this work proposes Feature-wise Linear Modulation for Cross-Problem Multi-Depot Vehicle Routing (FiLMMeD), the first MTL model explicitly targeting the MDVRP, to the best of our knowledge. Our main contributions are summarized as follows: • We introduce an MTL model capable of handling a wide range of MDVRP variants, including 8 new variants with inter-depot routes. In total, FiLMMeD can solve 24 distinct MDVRP variants, including backhauls, open routes, route length limits, time windows, inter-depot routes, and possible combinations between these constraints. • We introduce a feature-wise linear modulation (FiLM) mechanism (Perez et al., 2018) to condition node embeddings directly on the constraint attributes of each instance. FiLM is a conditioning technique that allows neural networks to adapt their internal representations based on auxiliary information. In our case, the conditioning input is a Boolean attribute vector indicating the presence or absence of each constraint.
A. Corrêa et al.
• To solve single-depot VRP variants with FiLMMeD, we fine-tune and train it using preference optimization (PO) instead of reinforcement learning (RL). PO is an emerging alternative to RL that has seen some applications to CO recently (Pan et al., 2025a; Liao et al., 2025). As far as we know, this is the first application of PO in an MTL setting, being an initial demonstration of its potential for future MTL works. Through various experiments, we show that PO significantly outperforms the traditional RL training of MTL models, having the potential to become the standard for future MTL works. • We propose a lightweight yet effective curriculum learning (CL) strategy for the MDVRP, which structures training by gradually increasing the number of active constraints, leading to improved generalization. We show that, unlike single-depot VRPs where uniformly sampling suffices, the MDVRP may induce more complex interactions between constraints, requiring a guided curriculum for effective generalization. We report extensive experimental results on 24 MDVRP variants and 16 single-depot VRP variants, with FiLMMeD achieving state-of-the-art performance and surpassing existing MTL baselines. Ablation studies confirm the effectiveness of the proposed FiLM mechanism, CL training regimen and PO algorithm. The rest of this paper is organized as follows. Section 2 discusses the relevant literature. Section 3 establishes important background about our work. Section 4 details the main architecture and methodology in this study. Section 5 displays all computational experiments performed. Finally, Section 6 draws conclusions and possible future work directions.
2. Related Work 2.1. Neural-based CO for VRPs Recently, neural-based methods have been growing as a promising third paradigm for CO problems, complementing (meta-)heuristics and exact algorithms (Bengio et al., 2021; Mazyavkina et al., 2021; Bogyrbayeva et al., 2024). Generally speaking, neural-based methods are, most of the time, construction-based, focusing on learning policies to incrementally build a solution in an end-to-end manner. One of the first prominent examples are Pointer Networks, a sequence-to-sequence method used to construct solutions autoregressively for the traveling salesman problem (TSP) (Vinyals et al., 2015). Pointer Networks marked an important paradigm shift in the literature, being one of the first demonstrations of the potential for neural-based CO solvers. Despite showing competitive performance on the TSP, it relied on supervised learning for training. Considering the NP-hard nature of many CO problems, computing (near)optimal labels is non-trivial, especially for larger instances. Thus, to circumvent this, following works focused on training with RL instead of supervised learning (Bello et al., Page 2 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
2017; Nazari et al., 2018). Later, Kool et al. (2019) made a substantial contribution to the literature, developing an attention-based model (AM) inspired by the Transformer architecture (Vaswani et al., 2017). The AM is, up to this day, the inspiration and backbone for most existing VRP model architectures in the literature. After, Kwon et al. (2020) introduced POMO, a new training regimen for neural-based models that exploits the symmetries inherent in CO problems. Their key contributions are twofold. First, they proposed REINFORCE with shared baselines, an algorithm that generates multiple trajectories per instance, leveraging different solution starting nodes to compute a low-variance baseline. This stabilizes training and yields substantially faster convergence than the conventional REINFORCE algorithm. Second, they developed a lightweight instance augmentation technique using reflections and rotations. Together, both techniques have become a standard across almost all neuralbased CO works, influencing most subsequent studies (Bi et al., 2025; Chalumeau et al., 2023; Corrêa et al., 2026; Drakulic et al., 2023; Grinsztajn et al., 2023; Jiang et al., 2026; Lei et al., 2022; Kim et al., 2022; Zhou et al., 2023, 2024b). Other less common approaches can be classified as improvement-based, which learn to iteratively refine an initially generated solution through local search operators (Chen and Tian, 2019; Hottung and Tierney, 2020; Ma et al., 2021; Hudson et al., 2022; Roberto et al., 2020; Wu et al., 2022; Ma et al., 2023). Typically, improvement-based methods find higher-quality solutions than construction-based, but at the cost of significantly slower inference. Lastly, divide-and-conquer approaches have also been showing promise lately, learning to solve VRPs by decomposing an instance into multiple smaller sub-instances (Cheng et al., 2023; Fu et al., 2021; Hou et al., 2023; Kim et al., 2021; Li et al., 2021; Ye et al., 2024; Zheng et al., 2024; Zong et al., 2022). However, these methods may suffer from errors at sub-problem boundaries and increased training and implementation complexity.
2.2. Multi-task learning for VRPs In recent years, the neural-based CO community has started to investigate the application of MTL to improve the generalization across different VRP variants. Liu et al. (2024) introduced MTPOMO, the first MTL-based approach for VRPs capable of solving 16 different variants through attribute composition. This work established a new paradigm for cross-problem learning, significantly influencing later MTL developments. After, Zhou et al. (2024a) proposed MVMoE, an approach built upon MTPOMO which incorporates mixture-of-experts in its architecture. In a mixtureof-experts model, the network is divided into multiple specialized sub-networks, called experts, each designed to learn distinct patterns or features. Then, a gating network selects the most relevant experts to be activated, depending on the given input. Another significant step forward is RouteFinder (Berto et al., 2025), which encompasses several technical
A. Corrêa et al.
innovations. RouteFinder adopts a modern Transformerbased architecture, including root mean square normalization (Zhang and Sennrich, 2019) and SwiGLU activations (Shazeer, 2020). It also employs mixed batch training, a technique that allows multiple different VRP variants to be processed simultaneously within the same batch, which leads to more stable training. Li et al. (2025) presented CaDA, incorporating a dual-attention mechanism that adds a new multi-head attention module to the encoder architecture, focused on capturing the information of closely related nodes. Specifically, this is achieved using top-k sparse attention, where each node only attends to its k most relevant neighbors. The aforementioned studies are considered as the backbone of MTL VRP research, inspiring most following works. Wang et al. (2025) introduced a soft-parameter-sharing model to balance task-specific and shared representations across different variants. Although improving performance over MTPOMO and MVMoE, its multiple separate MultiHead Attention modules introduce significant more computational burden and GPU memory footprint. Huang et al. (2025) presented ReLD, showing that adding a simple identity mapping with a feed-forward layer to the decoder can significantly enhance its capacity. ReLD surpasses stateof-the-art performance on single- and multi-task settings, providing a great lightweight alternative to expensive heavy decoder Transformer models. Goh et al. (2025) proposed SHIELD, improving robustness to both cross-problem and cross-distribution generalization through hierarchical and sparsity mechanisms. More recently, Pan et al. (2025b) proposed MoSES, which reformulates VRP variants through a State-Decomposable Markov Decision Process and employs specialized LoRA experts with adaptive gating. Although it achieves strong performance, MoSES requires training and maintaining multiple task-specific expert networks, substantially increasing computational cost and memory footprint compared to previous architectures. In an effort to model the highly varied topologies inherent to different VRP variants, Liu et al. (2025) introduced a novel pre-training framework based on mixed curvature. Rather than relying on a single geometric space, their approach maps nodes into a composite manifold that fuses Euclidean, spherical, and hyperbolic dimensions, providing the architectural flexibility required to capture the diverse underlying topological structures of different variants. Finally, Zheng et al. (2025) presented MTL-KD, leveraging knowledge distillation for a better cross-size generalization. Remark. Recent MTL advances have improved crossproblem generalization for neural VRP solvers, yet significant gaps remain. First, existing approaches focus almost exclusively on single-depot variants, leaving the multi-depot setting — critical for modern logistics — largely unaddressed. Second, while some works incorporate constraint information through static feature concatenation or promptbased conditioning, they fail to dynamically modulate internal representations based on active constraint combinations.
Page 3 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
Figure 1: MDVRP constraints addressed in our work.
This approach exacerbates gradient interference across diverse variants, restricting generalization. Third, prior training strategies (whether uniformly sampling single-constraint variants or simultaneous exposure to all variants), while effective on the single-depot setting, have shown limited generalization on the MDVRP. Lastly, the reliance on RLbased training in MTL settings limits convergence stability and solution quality, suffering from gradient interference and diminishing reward signals, as noted by Pan et al. (2025a). These limitations motivate our contributions, with our work diverging from previous efforts in multiple key aspects: we explicitly focus on the underexplored multidepot setting, which has received limited attention in neuralbased research; we introduce a lightweight FiLM mechanism to explicitly condition the learned representations on the active combination of problem constraints, showing that it effectively disentangles the latent space of node embeddings, leading to improved generalization; we design a dedicated CL training regimen for the multi-depot setting, which gradually increases the complexity of training by the number of active constraints in the variants sampled, outperforming prior training strategies; finally, we employ PO to fine-tune and train FiLMMeD model on single-depot variants, demonstrating that it significantly outperforms the conventional REINFORCE loss function, establishing it as a promising direction for future MTL works.
3. Preliminaries 3.1. Problem definition In this study, we solve 24 different MDVRP variants, whose formulations we describe in this subsection. A classic MDVRP instance is defined on a graph = {, }. Here, = {𝑣0 , ..., 𝑣𝑚−1 , 𝑣𝑚 , ..., 𝑣𝑚+𝑛−1 } is the set of nodes in the problem, where 𝑑 = {𝑣0 , ..., 𝑣𝑚−1 } denotes 𝑚 depots and 𝑐 = {𝑣𝑚 , ..., 𝑣𝑚+𝑛−1 } denotes 𝑛 customers. The edge set = {𝑒𝑖𝑗 ∶ 𝑖, 𝑗 ∈ , 𝑖 ≠ 𝑗, (𝑖, 𝑗) ∉ 𝑑 × 𝑑 } contains all connections between distinct nodes, excluding direct depotto-depot edges. Each edge is associated with a travel cost 𝑐𝑖𝑗 . At each depot, a fleet of vehicles is available, where each vehicle has a capacity limit 𝐶. A fundamental constraint across all MDVRP variants is that every customer 𝑣𝑖 has a demand 𝛿𝑖 , which must be satisfied by exactly one vehicle dispatched from one of the 𝑚 depots. The solution 𝜏 represents the sequence of nodes visited in the problem, and is composed A. Corrêa et al.
of multiple individual routes, each corresponding to the path followed by a single vehicle. Once a vehicle completes its route, it must return to the depot from which it departed. A solution is feasible if all customer demands are satisfied and no vehicle exceeds the capacity limit on its assigned route. The objective is to find the optimal solution 𝜏 ∗ that minimizes the total travel distance across all vehicles. Besides the classic MDVRP, we also solve other variants by adding constraints on top of the original problem formulation. In our work, we explore five additional constraints, four of which have been addressed in prior MTL works, but mostly focused on single-depot VRPs (Liu et al., 2024; Zhou et al., 2024a; Berto et al., 2025). We also consider another constraint specific to MDVRPs, which are inter-depot routes (Crevier et al., 2007; Ramos et al., 2020), leading to 8 additional MDVRP variants previously unsolved in neural-based works. A brief description of the five additional constraints is provided below, and an illustration is provided in Figure 1. Backhaul (B): In the standard MDVRP, a vehicle’s load decreases whenever it visits a customer. In the MDVRP with backhauls (MDVRPB), some customers require pickups instead of deliveries, causing the vehicle’s load to increase. These customers are called backhaul customers, whereas traditional delivery customers are linehaul customers. In this paper, we adopt two different backhaul settings. Across the experiments compared to Zhou et al. (2024a), we use the mixed backhaul, in which a single vehicle may alternate between serving linehaul and backhaul customers without any strict precedence order. For experiments compared to Berto et al. (2025), a single vehicle must not alternate between back and linehaul customers, i.e., it may only visit backhaul customers after completing all its linehaul visits. Route length limit (L): In the MDVRP with route length limits (MDVRPL), the length/duration of each vehicle’s route must not exceed a prescribed threshold 𝐷. Open route (O): In the MDVRP with open routes (MDOVRP), vehicles are not required to return to their departure depot after completing their respective routes. Therefore, each route ends at the last visited customer. Time window (TW): In the MDVRP with time windows (MDVRPTW), each node 𝑣𝑖 ∈ has an associated time window [𝑒𝑖 , 𝑙𝑖 ] and a service time 𝑠𝑖 . A vehicle visiting node 𝑣𝑖 must arrive no later than 𝑙𝑖 . Service can only begin within the time window [𝑒𝑖 , 𝑙𝑖 ], requiring a wait if the arrival is
Page 4 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP Table 1 24 MDVRP variants with five constraints. Open Route (O) MDVRP MDOVRP MDVRPB MDVRPL MDVRPTW MDOVRPTW MDOVRPB MDOVRPL MDVRPBL MDVRPBTW MDVRPLTW MDOVRPBL MDOVRPBTW MDOVRPLTW MDVRPBLTW MDOVRPBLTW MDVRPI MDVRPIB MDVRPIL MDVRPITW MDVRPIBL MDVRPIBTW MDVRPILTW MDVRPIBLTW
Backhaul (B)
Duration Limit (L)
Time Window (TW)
Inter-depot routes (I)
✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓
✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓
✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
earlier than 𝑒𝑖 . All depots are also assigned an ending time window, requiring vehicles to arrive at the depot before it. Inter-depot routes (I): The MDVRP with inter-depot routes (MDVRPI) is a relatively less-studied variant of the MDVRP. In this setting, vehicles are allowed to stop at any depot during a route to reload their capacity 𝐶 (hence the term inter-depot routes). However, each vehicle must still return to its original departing depot at the end of its route. Consistent with the general problem definition, direct depotto-depot edges are excluded. The original MDVRPI formulation also includes route length limit constraints (Crevier et al., 2007; Ramos et al., 2020). However, in our work, we relax this requirement to allow for a broader set of MDVRPI variants. Nonetheless, some of the MDVRPI variants we consider do include route length limits (see Table 1). In total, we solve eight MDVRPI variants, whereas only four would be possible without relaxing the route length limit constraint. The combination of the aforementioned constraints lead to different MDVRP variants, which we detail in Table 1.
3.2. Learning to solve MDVRPs Most neural-based models generate solutions autoregressively by learning a policy 𝜋𝜃 through an attention-based neural network (Kool et al., 2019; Kwon et al., 2020; Liu et al., 2024; Zhou et al., 2024a). We adopt this approach, utilizing an encoder-decoder framework, where a complete solution is built timestep by timestep by framing the MDVRP as a Markov decision process (MDP). The process begins with the encoder embedding an input instance based on its underlying features, such as node coordinates, demands and time windows. A decoder then constructs the solution autoregressively. At each timestep 𝑡, it receives a state, from which it computes the probability of selecting each candidate node next to extend the partial solution. Similar to prior neural MDVRP models (Li et al., 2024; Corrêa et al., 2026), FiLMMeD constructs solutions by building one route at a time. To encourage systematic exploration of the solution space, the initial action is chosen deterministically following the POMO framework (Kwon et al., A. Corrêa et al.
2020). For example, in an instance with 50 customers and 3 depots, there may be up to 150 possible customer–depot initializations, resulting in 150 distinct trajectories (i.e., solutions) evaluated in parallel. After that, the model sequentially constructs the current route by selecting the next customer to be served, until it decides to end the route, which occurs when the agent selects the departure depot. Upon terminating the route, the agent selects the depot from where the next route will begin (which can be the same depot or not). The process repeats until a full solution is constructed. This decision-making sequence can be formalized as an MDP, whose individual components we describe below: State: At each timestep 𝑡, the agent observes a state 𝑠𝑡 , which represents the current partial MDVRP solution. The state comprises the embeddings of the previously selected node together with state variables carrying information about the partial solution. Action: After observing the state 𝑠𝑡 , the agent selects an action 𝑎𝑡 ∈ 𝑡 , where 𝑡 denotes the set of feasible actions at timestep 𝑡. Here, 𝑎𝑡 can be of two types, as discussed before: 1) a depot node, which is chosen to either start a new route or terminate the current one. In the case of variants with inter-depot routes, a depot node (one that is different from the starting depot) can be chosen throughout a route to replenish the vehicle’s capacity; 2) a customer node, which represents the next node to be visited by the vehicle in the current route. State transition: After executing 𝑎𝑡 , the environment transitions to a new state 𝑠𝑡+1 reflecting the updated partial solution. The vehicle’s capacity 𝐶𝑡 , elapsed time 𝑇𝑡 and route length 𝐷𝑡 are updated accordingly, and its location is set to the coordinates of the selected node. The feasibility mask is then recomputed to exclude visited nodes, as well as any unvisited nodes that are no longer reachable due to the tighter constraints induced by the chosen action. Reward: The reward 𝑟(𝜏, ) is computed after an entire solution 𝜏 is constructed for instance . It is defined as the negative sign of the total distance traveled by all vehicles, meaning that maximizing the expected reward corresponds to minimizing the total travel distance. Policy: Throughout training, the agent learns a decisionmaking policy 𝜋𝜃 parameterized by an attention-based neural network. At each timestep 𝑡, given the current state 𝑠𝑡 , the policy outputs a probability distribution over feasible next nodes, guiding the choice of action 𝑎𝑡 . Then, the agent chooses the next action greedily or by stochastic sampling, according to this distribution.
4. Methodology In this section, we present the architecture of our approach and important methodological details. We first present an overview of our model, including the FiLM mechanism and the rationale behind it. Then, we present the components of our training regimen, including the CL strategy and the PO algorithm. Page 5 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
4.1. FiLMMeD architecture FiLMMeD follows the general encoder-decoder architecture of previous Transformer-based MTL models (Liu et al., 2024; Zhou et al., 2024a). In this work, we apply FiLMMeD modifications into three base architectures: FiLMMeD-MTPOMO, FiLMMeD-MVMoE, and FiLMMeDCaDA. To streamline the description, we primarily use MTPOMO and MVMoE as references since they share a similar model pipeline. For specific architectural details regarding the FiLMMeD-CaDA model, we refer readers to Li et al. (2025). We note, however, that the FiLM mechanism is integrated into CaDA in the exact same manner described for the other models. Furthermore, our core contributions remain modular, i.e., they can be seamlessly adapted to a wide variety of other neural-based MTL architectures.
4.1.1. Encoder The encoder starts by processing a given MDVRP instance . First, linear layers are used to project the static features of each node in the instance onto a 𝑑-dimensional embedding space. Each depot node 𝑣𝑗 ∈ 𝑑 , ∀𝑗 ∈ {0, ..., 𝑚− 1} is characterized by its two-dimensional coordinates. The features of each customer node 𝑣𝑖 ∈ 𝑐 , ∀𝑖 ∈ {𝑚, ..., 𝑚 + 𝑛 − 1} include the two-dimensional coordinates, demand, early and late time windows, and service time. Let ℎ(𝑑) 𝑗 and ℎ(𝑐) 𝑖 denote the initial embeddings of depot and customers nodes, respectively. First, two separate linear layers are used to process each set of features from depots and customers, as follows: (𝑑) ℎ(𝑑) 𝑗 = 𝑊0 𝑥𝑣𝑗 + 𝑏0 , ∀𝑣𝑗 ∈ 𝑑
(1)
(𝑐) ℎ(𝑐) 𝑖 = 𝑊1 𝑥𝑣 + 𝑏1 , ∀𝑣𝑖 ∈ 𝑐
(2)
𝑖
2 where 𝑥(𝑑) 𝑣𝑗 ∈ ℝ denotes the feature vector of depot 𝑣𝑗 , 6 𝑥(𝑐) 𝑣𝑖 ∈ ℝ denotes the feature vector of customer 𝑣𝑖 , 𝑊0 ∈ 𝑑×2 ℝ and 𝑊1 ∈ ℝ𝑑×6 are learnable weight matrices, and 𝑏0 , 𝑏1 ∈ ℝ𝑑 are bias vectors. After that, we modulate the customer embeddings using FiLM (Perez et al., 2018). FiLM is a conditioning technique that allows a neural network to conditionally modify an embedding based on external information. Here, each customer embedding ℎ(𝑐) 𝑖 is explicitly modulated according to a Boolean conditioning vector 𝑧 ∈ ℝ5 , which indicates the active constraints in the input instance, that is, whether an instance contains B, L, O, TW or I attributes. More specifically, FiLM computes feature-wise scaling (𝛾) and feature-wise shifting (𝛽) parameters based on the conditioning vector:
𝛾 = 𝑓𝛾 (𝑧) ∈ ℝ𝑑 ,
𝛽 = 𝑓𝛽 (𝑧) ∈ ℝ𝑑
(3)
where 𝑓𝛾 and 𝑓𝛽 are learnable linear projections. Each customer embedding is then modulated as an affine transformation: (𝑐) 𝐹 𝑖𝐿𝑀(ℎ(𝑐) 𝑖 |𝛾, 𝛽) = 𝛾ℎ𝑖 + 𝛽.
A. Corrêa et al.
(4)
Here, 𝛾 serves a scaling parameter that controls the importance of each node in response to the conditioning vector. For example, if time windows are active, 𝛾 might emphasize the embedding channels related to early and late time windows, making the model more sensitive to time windows constraints during subsequent transformations. 𝛽 acts as a bias parameter that shifts each embedding by an offset determined by the active set of constraints. This allows the model to adapt the learned representations of each node based on its context, instead of treating all embeddings identically regardless of the variant solved. After that, the result embeddings are concatenated with the previously processed depot embeddings, resulting in ℎ(0) 𝑖 , ∀𝑣𝑖 ∈ . This sequence is then processed by 𝐿 encoder layers. Each layer 𝓁 ∈ {1, … , 𝐿} consists of an MHA sub-layer followed by a node-wise Feed-Forward Network (FFN), with residual connections and Instance Normalization (IN) before and after the FFN. In the MHA sub-layer with 𝐴 heads, for each node 𝑣𝑖 and head 𝑎 ∈ {1, … , 𝐴}, we compute query (𝑄), key (𝐾), and value (𝑉 ) vectors: 𝑄(𝓁,𝑎) = 𝑊𝑄(𝓁,𝑎) ℎ(𝓁−1) 𝑖 𝑖
(5)
𝐾𝑖(𝓁,𝑎) = 𝑊𝐾(𝓁,𝑎) ℎ(𝓁−1) 𝑖
(6)
𝑉𝑖(𝓁,𝑎) = 𝑊𝑉(𝓁,𝑎) ℎ(𝓁−1) 𝑖
(7)
where 𝑊𝑄(𝓁,𝑎) , 𝑊𝐾(𝓁,𝑎) , 𝑊 (𝓁,𝑎) 𝑉 ∈ ℝ𝑑𝑘 ×𝑑 are learnable projection matrices, and 𝑑𝑘 = 𝑑∕𝐴. The attention weight between nodes 𝑣𝑖 and 𝑣𝑗 is computed as: ⎛ (𝑄(𝓁,𝑎) )⊤ 𝐾 (𝓁,𝑎) ⎞ 𝑖 𝑗 ⎟. √ ⎟ 𝑑 𝑘 ⎠ ⎝
𝑤(𝓁,𝑎) = Softmax ⎜ 𝑖𝑗 ⎜
(8)
This weight represents the influence of node 𝑣𝑗 on node 𝑣𝑖 within the 𝑎-th head. The output of head 𝑎 for node 𝑣𝑖 is ∑ a weighted sum of values: 𝑜(𝓁,𝑎) = 𝑣𝑘 ∈ 𝑤(𝓁,𝑎) 𝑉𝑘(𝓁,𝑎) . The 𝑖 𝑖𝑘 outputs from all heads are concatenated and projected: [ ] MHA(ℎ(𝓁−1) ) = 𝑊𝑂 𝑜(𝓁,1) , … , 𝑜(𝓁,𝐴) (9) 𝑖 𝑖 𝑖 where 𝑊𝑂 ∈ ℝ𝑑×𝑑 . The MHA output is then processed by the first residual and normalization block: ( ) (𝓁−1) (𝓁−1) ℎ̃ (𝓁) = IN ℎ + MHA(ℎ ) (10) 𝑖 𝑖 𝑖 Finally, the standard FFN is applied, followed by a second residual and normalization block to produce the layer output: ( ) ̃ (𝓁) + FFN(ℎ̃ (𝓁) ) ℎ(𝓁) = IN ℎ (11) 𝑖 𝑖 𝑖 Page 6 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
The FFN consists of two linear transformations with a ReLU activation in between. The description above details the encoder of FiLMMeD-MTPOMO. For FiLMMeD-MVMoE, the only change is that the standard FFN layer is replaced by a mixture-of-experts layer (Zhou et al., 2024a).
4.1.2. Decoder After the encoding process, the decoder operates on 𝑚+𝑛−1 the final node embeddings {ℎ(𝐿) produced by the 𝑖 }𝑖=0 encoder. Like prior MTL architectures, we adopt a heavy encoder and light decoder structure, consisting of 6 encoder layers and a single decoder layer (Liu et al., 2024; Zhou et al., 2024a; Berto et al., 2025). Since the encoder runs only once per instance and the decoder relies on a single MHA module for repeated steps, this design ensures computational efficiency. First, the decoder utilizes its own set of learnable projections to compute keys and values based on the encoded node embeddings, which remain static throughout the decoding process: 𝐾𝑖𝑑𝑒𝑐 = 𝑊𝐾𝑑𝑒𝑐 ℎ(𝐿) 𝑖 ,
𝑉𝑖𝑑𝑒𝑐 = 𝑊𝑉𝑑𝑒𝑐 ℎ(𝐿) 𝑖 ,
∀𝑣𝑖 ∈ (12)
where 𝑊𝐾𝑑𝑒𝑐 , 𝑊𝑉𝑑𝑒𝑐 ∈ ℝ𝑑×𝑑 . At each decoding step 𝑡, a context feature vector is constructed from the embedding of the previously selected node and the current state variables. The state variables include the vehicle’s remaining capacity, the elapsed time, current route length, and Boolean indicators on the presence of open and inter-depot routes. This vector is projected to form the dynamic query 𝑄𝑑𝑒𝑐 : (𝑡) 𝑄𝑑𝑒𝑐 = 𝑊𝑄𝑑𝑒𝑐 Concat(ℎ(𝐿) 𝜏 , 𝐶𝑡 , 𝑇𝑡 , 𝐷𝑡 , 𝑜𝑡 , 𝑖𝑑𝑡 ), (𝑡) 𝑡−1
(13)
where ℎ(𝐿) 𝜏𝑡−1 denotes the embedding of the previously selected node. Then, an MHA mechanism is employed. The query 𝑄𝑑𝑒𝑐 (𝑡)
attends to the static keys 𝐾𝑖𝑑𝑒𝑐 to aggregate the values 𝑉𝑖𝑑𝑒𝑐 , ′ producing an updated context vector ℎ𝑐𝑡 . Any infeasible node 𝑣𝑗 has its attention score set to −∞. Finally, we compute the compatibility score 𝑢𝑖 for each node 𝑣𝑖 using a single-head attention mechanism. This step uses the updated context ′ vector ℎ𝑐𝑡 as the query and the original encoder embeddings ℎ(𝐿) 𝑖 as the keys: ( ′ 𝑡 𝑇 (𝐿) ) ⎧ (ℎ𝑐 ) ℎ𝑖 √ , ⎪𝜉 ⋅ tanh 𝑑𝑘 𝑢𝑖 = ⎨ ⎪−∞, ⎩
if 𝑣𝑖 ∈ 𝑡 , otherwise.
(14)
Here, 𝑑𝑘 = 𝑑∕𝐴 (with 𝐴 being the number of heads), and 𝜉 is a clipping parameter set to 10 (Bello et al., 2017). Similar to the MHA step, we mask the compatibilities of infeasible nodes by setting 𝑢𝑖 = −∞. Finally, the probability of selecting node 𝑣𝑖 at step 𝑡 is computed via the Softmax function: 𝑒𝑢𝑖 𝑝𝑖 = Softmax(𝑢𝑖 ) = ∑ 𝑢 𝑗 𝑗𝑒 A. Corrêa et al.
(15)
During inference, the nodes are always selected greedily (that is, argmax𝑖 𝑝𝑖 ), while during training, they are selected by sampling from the generated distribution.
4.2. Theoretical motivation behind FiLM Next, we provide theoretical arguments for why FiLM conditioning may improve policy learning and generalization across heterogeneous variants. Our analysis centers across two key aspects: 1) the gradient conflicts problem in MTL; 2) compositional generalization. Gradient conflict mitigation: When training a unified model across multiple (MD)VRP variants, the primary challenge lies in gradient interference (Yu et al., 2020). Essentially, different tasks (in our case, different constraints) induce distinct loss landscapes. For instance,the gradients for variants with time windows, backhauls and open routes generally exhibit a low cosine similarity. Without proper conditioning, gradients from variant 𝐴 may destructively interfere with those from variant 𝐵, leading to performance degradation on both, or one of them performing better in detriment of the other. Formally, let denote the set of variants and 𝜒 (𝜃) the loss for variant 𝜒 ∈ . Standard MTL suffers from conflicting updates when ⟨∇𝜃 𝜒1 , ∇𝜃 𝜒2 ⟩ < 0 for 𝜒1 ≠ 𝜒2 . FiLM can alleviate this issue by introducing task-specific affine modulations (𝛾𝜒 , 𝛽𝜒 ) applied to node’ representations. These modulations condition the forward pass, allowing different variants to induce distinct feature transformations before gradients are computed. As a result, variant-specific signals can be partially absorbed by the FiLM parameters, reducing the pressure on shared weights to accommodate conflicting objectives. While this mechanism does not eliminate gradient conflicts in the shared parameters, it provides a lightweight pathway for task-specific adaptation, enhancing stability and generalization without fully separating the model into distinct experts, which is computationally prohibitive. Compositional generalization: Another theoretical advantage of FiLM is its ability to generalize to unseen constraint combinations. By factorizing the conditioning vector 𝑧 into independent constraint indicators, the model learns constraint-specific transformations that compose via superposition. This is similar to what hypernetworks do (Ha et al., 2017), where the adaptation parameters for a new task are generated by a meta-network rather than learned directly. When encountering a novel combination for the first time, the model can approximate the optimal transformation by interpolating between the learned transformations for each active constraint.
4.3. Training 4.3.1. Curriculum learning CL is a training strategy in which the model is exposed to tasks of progressively increasing complexity. By first learning to solve simpler problems, the model acquires fundamental patterns that can later be extended and refined as the task complexity increases. In the context of our work, Page 7 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
complexity is determined by the number and combination of active constraints in each MDVRP instance. We implement CL by progressively expanding the set of variants from which the model learns during training. In the beginning, the curriculum is dominated by single-constraint problems, allowing the model to learn the effect of each constraint in isolation. Concretely, during the first 30% of epochs, each batch is sampled from one of the following variants: MDVRP, MDOVRP, MDVRPB, MDVRPL, MDVRPTW, MDVRPI, MDOVRPTW, MDVRPBTW, MDVRPITW. While most of the variants available during the initial phase of the curriculum are single-constraint, we also include three two-constraint variants with time windows (MDOVRPTW, MDVRPBTW, and MDVRPITW). This choice is not arbitrary, but motivated by empirical observations specific to the multi-depot setting: learning solely from variants with a single constraint (like MTPOMO (Liu et al., 2024)), led to poor convergence on several MDVRP variants. In particular, variants involving backhauls, open routes, or inter-depot routes showed a very slow convergence, and was further exacerbated when paired with time windows, which strongly interact with other constraints. While typical training strategies have proven effective for single-depot cases (Zhou et al., 2024a; Berto et al., 2025), we found they achieved sub-par generalization for the MDVRP. More discussion on this issue is provided in Section 5.4.1. After this initial phase, we expand the curriculum, with all remaining two-constraint variants being introduced next. At 60% of total training epochs, we add the full set of three-constraint variants, and finally, at 90%, the remaining ones. As demonstrated in our ablation studies, this strategy proved to improve convergence, and ultimately, generalization across the tested MDVRP variants. To train our model, we employ the REINFORCE algorithm with shared baselines (Kwon et al., 2020), an RL method designed specifically for CO problems. A description of this algorithm is provided next.
4.3.2. REINFORCE with shared baselines To train the MDVRP models of FiLMMeD, like most neural-based VRP works, we employ the REINFORCE with shared baselines algorithm (Kwon et al., 2020). Each epoch comprises 100,000 instances, which are sampled in batches of size 𝐵. For each instance 𝑖 , ∀𝑖 ∈ {1, ..., 𝐵}, the decoder generates 𝑁 trajectories {𝜏𝑖1 , … , 𝜏𝑖𝑁 }. This is achieved by enforcing diverse starting nodes for each trajectory, allowing the model to broadly explore the solution space in a computationally efficient manner. By generating multiple parallel rollouts, a low-variance baseline 𝑏𝑖 can be calculated for each instance 𝑖 , as the average reward of all its 𝑁 generated trajectories: 1 ∑ 𝑅(𝜏𝑖𝑗 , 𝑖 ). 𝑁 𝑗=1 𝑁
𝑏𝑖 =
A. Corrêa et al.
(16)
Then, the policy gradient ∇𝜃 𝐽 (𝜃) is approximated using the advantage of each trajectory relative to this shared baseline: ∇𝜃 𝐽 (𝜃) ≈
𝐵 𝑁 ( ) 1 ∑∑ 𝑅(𝜏𝑖𝑗 , 𝑖 ) − 𝑏𝑖 ∇𝜃 log 𝑝𝜃 (𝜏𝑖𝑗 |𝑖 ). 𝐵𝑁 𝑖=1 𝑗=1
(17) Finally, the model parameters 𝜃 are updated via gradient ascent, with a learning rate 𝜂. During training, we set the number of parallel trajectories to 𝑁 = 𝑚 + 𝑛 − 1 (Li et al., 2024; Corrêa et al., 2026). This results in 𝑁 = 52 for 50-node instances and 𝑁 = 102 for 100-node instances, since we consider 3 depots. The set of starting nodes includes the last two depots (i.e., depots with indexes 1 and 2) and all available customers. For trajectories starting at a customer node, the vehicle is assumed to depart from the first depot (depot with index 0). This approach allows us to cover a wide range of initial solutions without incurring the computational burden of enumerating all possible starting nodes (3 × 50 = 150). We note that this choice does not impose any structural biases on the final solution, since the model can choose any other depot to anchor subsequent routes during the decoding process.
4.3.3. Preference optimization Although FiLMMeD is primarily designed for MDVRP variants, we do not wish to neglect single-depot VRPs, which constitute a large portion of practical routing problems and remain widely used benchmarks. Therefore, we fine-tune the MDVRP-pre-trained model on 16 single-depot VRPs previously explored in MTL research (Zhou et al., 2024a). Importantly, we employ PO instead of RL for this fine-tuning stage. We also train additional single-depot models from scratch using PO, under the same experimental settings as Berto et al. (2025). Our choice for pre-training MDVRP models using REINFORCE rather than PO was to ensure a fairer assessment against established baselines – namely, REINFORCE itself and the uniform single-constraint sampling, opposed to our CL strategy. This baseline comparison allowed us to validate FiLMMeD’s architectural innovations, that is, FiLM and CL, without introducing confounding effects from a different training paradigm. Nevertheless, we did perform ablation studies on pre-training MDVRP models with PO, which significantly outperformed its REINFORCE counterpart (see Section 5.4.2). Besides solving single-depot problems, this design choice serves a broader purpose. While PO has recently shown improved generalization and sample efficiency (Pan et al., 2025a), its application has been largely limited to non-MTL domains. Our results suggest that PO is not merely a viable alternative to REINFORCE, but a more principled training paradigm for MTL, addressing several of its fundamental limitations. We also demonstrated its broad applicability across a variety of different model architectures, with PO outperforming RL on every occasion (see Section 5.8). Page 8 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
In summary, PO replaces typical numeric rewards with pairwise preferences between candidate solutions. Instead of learning from scalar rewards, the model is optimized from qualitative comparisons, that is, whether one solution is better than another according to a given objective. This approach stabilizes learning, mitigating issues related to diminishing reward signals and promoting a better exploration of the solution search space. Concretely, at each batch, we randomly select 𝐵 instances from one of the 16 single-depot VRP variants (see Table 2 for a full list of variants). For each instance 𝑖 , ∀𝑖 ∈ {1, ..., 𝐵}, we generate 𝑁 solutions (setting again 𝑁)𝑚+ 𝑛− 1) and compute their ground-truth rewards 𝑟𝑖1 , ..., 𝑟𝑖𝑁 . These rewards induce conflict-free preference labels, that is, solution 𝑗 is preferred over solution 𝑘 whenever 𝑟𝑗 > 𝑟𝑘 . Within each instance, we compute all possible preference label pairs by comparing every solution against every other one. Here, 𝑦𝑖𝑗,𝑘 denotes the preference label between solutions 𝑗 and 𝑘 which is equal to 1 if 𝑟𝑖𝑗 > 𝑟𝑖𝑘 , and 0 if not. For each solution 𝑗, the log-probabilities of the selected actions across the trajectory are accumulated, giving log𝜋𝜃 (𝜏𝑗𝑖 ). The difference of log-probabilities between solutions 𝑗 and 𝑘 is then mapped into a preference probability: 𝑝𝜃 (𝜏𝑗𝑖 ≻ 𝜏𝑘𝑖 ) = 𝜎(𝛼(log𝜋𝜃 (𝜏𝑗𝑖 ) − log𝜋𝜃 (𝜏𝑘𝑖 )))
(18)
where 𝛼 is an entropy regularization parameter and 𝜎 is a Sigmoid function. The PO loss is then defined as the negative log-likelihood of the observed preferences within all instances and pairs: PO = −
𝐵 𝑁 1 ∑ ∑ 𝑖 𝑦 log𝑝𝜃 (𝜏𝑗𝑖 ≻ 𝜏𝑘𝑖 ) 𝐵𝑁 2 𝑖=1 𝑗=1, 𝑘=1 𝑗,𝑘
(19)
Intuitively, this loss pushes the policy to assign greater probabilities to preferred trajectories relative to other solutions within the same instance. Our implementation follows the same algorithm proposed by Pan et al. (2025a), adapted for the MTL setting. Why PO for Multi-task Learning? One of the most crucial challenges for the application of MTL in CO lies in the gradient interference across different tasks. Simply put, different VRP variants induce conflicting parameter updates due to heterogeneous constraint structures. In this regard, we identify that PO holds several theoretical advantages over RL:
Variance reduction and scale invariance: The REIN-
FORCE with shared baselines algorithm, commonly used in neural-based CO works, relies on the gradient estimator ∇𝜃 𝐽 = 𝔼[(𝑅(𝜏, ) − 𝑏)∇𝜃 log 𝑝𝜃 (𝜏, )], where variance scales with the absolute variance of tour lengths across instances. In MTL, this is especially problematic, since different tasks typically have much different reward scales. For instance, the average costs for variants with time windows are generally much larger than variants without, causing high-cost instances to dominate gradients and implicitly A. Corrêa et al.
Table 2 16 VRP variants with four constraints. Open Route (O) CVRP OVRP VRPB VRPL VRPTW OVRPTW OVRPB OVRPL VRPBL VRPBTW VRPLTW OVRPBL OVRPBTW OVRPLTW VRPBLTW OVRPBLTW
Backhaul (B)
Duration Limit (L)
Time Window (TW)
✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓
✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓
prioritize certain tasks over others. In contrast, PO optimizes the Bradley-Terry objective PO , which depends only on relative tour rankings within a single instance. Since preferences 𝟙[𝑅(𝜏𝑗 , ) > 𝑅(𝜏𝑘 , )] are strictly invariant to reward scaling, PO ensures a much more stable learning. We verified this empirically by measuring gradients across multiple training batches. PO reduced per-parameter gradient variance by over 2000 times (from 2.33 × 10−6 to 1.09 × 10−9 ) and overall gradient magnitude by 72 times (from 1.00 to 1.39 × 10−2 ), while maintaining comparable signal-to-noise ratios (∼ 1478 vs 971). PO’s lower gradient magnitude may enable a more tolerant learning process with respect to gradient interference, as smaller updates mitigate conflicting tasks from violently overwriting each other’s parameters.
𝑂(𝑁 2 ) supervision density: For each instance 𝑖 , REINFORCE extracts 𝑁 scalar signals {𝑅(𝜏𝑖𝑗 , 𝑖 ) − 𝑏𝑖 }𝑁 , 𝑗=1 discarding information when multiple trajectories are nearoptimal and the policy converges (i.e., advantages ≈ 0). ( ) Conversely, PO computes 𝑁2 pairwise comparisons from each instance, providing a much denser credit assignment than REINFORCE, allowing it to extract more information from the same trajectories, leading to a faster convergence and lower performance gaps.
5. Experiments In this section, we validate the effectiveness of the proposed method through a series of experiments. We evaluate FiLMMeD on 24 MDVRP variants and 16 single-depot VRPs. We also conduct various ablation studies to measure the impact of all proposed contributions. Baselines: MDVRP variants: For traditional solvers, we use PyVRP (Wouda et al., 2024), a highly versatile hybrid genetic search (HGS) algorithm capable of solving all 24 MDVRP variants. We solve instances with 50 and 100 nodes, running PyVRP for 20 and 40 seconds per instance, respectively. Additionally, we report results using ten times the computational budget (200 and 400 seconds Page 9 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
per instance). Following prior work, we parallelized PyVRP across 32 CPU cores (Zhou et al., 2024a). All PyVRP default hyper-parameters were kept. For neural-based methods, we compare our method against MTPOMO (Liu et al., 2024) and MVMoE (Zhou et al., 2024a). We consider two versions of FiLMMeD, which incorporate our proposed contributions on top of MTPOMO and MVMoE, labeled as FiLMMeD-MTPOMO and FiLMMeD-MVMoE in the results. VRP variants: For single-depot VRPs, we evaluate FiLMMeD against two different experimental configurations: 1) those defined by Zhou et al. (2024a) (FiLMMeDMTPOMO and FiLMMeD-MVMoE); 2) those defined by Berto et al. (2025) (labeled as FiLMMeD-CaDA, incorporating our contributions on top of CaDA). Training: Each neural-based model was trained for 300 epochs, with each epoch consisting of 100,000 MDVRP instances generated on the fly. Separate models were trained for 50-node and 100-node instances, with a batch size of 128 and 64 for each instance size. Each batch contains instances from a randomly selected MDVRP instance. MTPOMO and MVMoE were trained with the same hyper-parameters as in their original works, while the remaining hyper-parameters for FiLMMeD-MTPOMO and FiLMMeD-MVMoE are listed in Table 3. For the fine-tuning phase (applicable only to single-depot VRPs following the Zhou et al. (2024a) experimentel setting), the FiLMMeD models pre-trained on MDVRP variants were further trained for 300 epochs using the PO algorithm described in Section 4.3.3 (with a random single-depot VRP variant sampled in each batch). During this phase, we observed no significant difference in the computational time per epoch compared to the traditional RL-based training. In contrast, for the Berto et al. (2025) experimental setting, we trained FiLMMeD-CaDA from scratch for 300 epochs, using the same hyper-parameters and configurations as Li et al. (2025). Unlike FiLMMeD-MTPOMO and FiLMMeDMVMoE, FiLMMeD-CaDA was not fine-tuned from a pretrained MDVRP model, since the models reported by Zhou et al. (2024a) were trained on 100M instances, while the ones evaluated under the settings of Berto et al. (2025) were trained on only 30M instances. This ensures a fairer comparison. Had we fine-tuned FiLMMeD-CaDA from a pre-trained MDVRP model (which itself was trained over 30M instances), the total training exposure (60M instances) would have been double that of the baselines. Inference: Across all neural-based models, we employed a greedy decoding combined with the multiple starting nodes strategy and the ×8 instance augmentation technique proposed by (Kwon et al., 2020). For each MDVRP variant, we generated two testing datasets, each containing 1,000 instances with 50 and 100 nodes and 3 depots (see Section 5.1 for the instance generation procedure). For single-depot VRPs in the Zhou et al. (2024a) setting (Table 5), we used the same testing datasets as in (Zhou et al., 2024a). For the Berto et al. (2025) setting (Table 9), we followed their respective experimental protocols. On singledepot problems, we used 50 and 100 starting nodes for A. Corrêa et al.
the 50- and 100-node instances, respectively. On MDVRP variants, we increased this to 150 and 300 starting nodes to enumerate all possible initial depot-customer assignments (given that each instance contains 3 depots). We report the average objective values across each dataset (Obj.), the average gap (Gap) relative to the best non-neural baseline, and the computational time (Time) required to solve all 1,000 instances. The best performing non-neural baseline is indicated with *. Hardware: Non-neural baselines were run on a machine with 32 GB of RAM and an Intel Core i9-13900. Neural-based models were executed on a machine with 45GB of RAM, an Intel Xeon Gold 5315Y and an Nvidia RTX A4000. We note that we compare some results against those reported in prior works (Zhou et al., 2024a; Li et al., 2025), which utilized faster GPUs. This hardware asymmetry explains some inference time discrepancies, and not architectural overhead introduced by our contributions (see Section 5.9 for more details).
5.1. Instance generation In this section, we describe the data generation process we followed for the MDVRP instances of each constraint type. Across all variants, the node coordinates for both customers and depots are randomly sampled from a uniform distribution 𝑈 (0, 1). We recall that for single-depot VRPs, we used the same datasets and instance generation processes as Zhou et al. (2024a) and Berto et al. (2025). Vehicle capacity (C): Vehicle capacity is a constraint implicitly present in all variants considered in this work; therefore, we do not list it as a separate constraint type in Tables 1 and 2. For this constraint, we adopt the standard setup used in prior routing works (Kool et al., 2019; Zhou et al., 2024a). The demand 𝛿𝑖 for each customer node 𝑣𝑖 ∈ 𝑐 is an integer sampled from the uniform distribution 𝑈 (1, 2, ..., 9). The vehicle capacity 𝐶 is set to 40, for instances with 𝑛 = 50, and to 50 for instances with 𝑛 = 100. Finally, the demand of each node is normalized by dividing it by 𝐶. The masking mechanism prevents the model from selecting nodes whose demand exceeds the vehicle’s remaining capacity and also masks out nodes that have already been served. Open route (O): This constraint does not require any additional data to be generated. When active, it is represented simply as a Boolean flag 𝑜𝑡 in the context vector. For instances that simultaneously include both open routes and route length limit constraints, we follow the standard convention: because vehicles are not required to return to their departure depots, the final leg from the vehicle’s last visited customer back to the depot is excluded from the route-length calculation. Likewise, for instances with both open routes and time windows, the depot’s closing time 𝑙𝑖 imposes no restrictions to vehicles, since they are not required to return. We do not combine open routes with inter-depot routes in our MDVRP variants. The reason is that one would essentially contradict the other. Inter-depot routes allow vehicles to return to other depots to replenish their original capacity, whereas open routes explicitly remove the Page 10 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP Table 3 Hyperparameters used by FiLMMeD-MTPOMO FiLMMeD-MVMoE. Hyperparameter Value Model Embedding dimension 128 𝑑ℎ Number of attention 8 heads 𝐴 Number of encoder lay- 6 ers 𝐿 Feedforward hidden di- 512 mension 𝑑𝑎 Tanh clipping 𝜉 10.0 Training Batch size Instances per epoch Optimizer Learning rate (LR) Weight decay Training epochs Fine-tuning Batch size Learning rate Weight decay LR scheduler LR milestones LR gamma Number of fine-tuning epochs Optimizer 𝛼 Inference Number of POMO starting nodes (MDVRP) Number of POMO starting nodes (VRP) Instance augmentation
and
128 (for 𝑛 = 50) and 64 (for 𝑛 = 100) 100,000 Adam 1 × 10−4 1 × 10−6 300 128 (for 𝑛 = 50) and 64 (for 𝑛 = 100) 3 × 10−4 1 × 10−6 MultiStepLR [270, 295] 0.1 300 Adam 0.03 150 for 𝑛 = 50 and 300 for 𝑛 = 100 50 for 𝑛 = 50 and 100 for 𝑛 = 100 ×8, following Kwon et al. Kwon et al. (2020)
requirement to return to the depots at the end of service. For this reason, we treat these two as mutually exclusive. Route length limit (L): To generate route length limits, we follow an approach similar to Berto et al. (2025). We begin by computing the Euclidean distance between every depot-customer pair. The largest of these distances provides a natural lower bound on the maximum route length 𝐷, ensuring that any vehicle can reach any customer from its depot and return. Doubling this value produces a lowerbound estimate on the distance a vehicle would need to travel to visit that customer and return to its depot. We then sample 𝐷 from a uniform distribution between this value and 3.0. During decoding, the masking mechanism restricts vehicles to serving only those customers for which the current route length – plus the cost of visiting the customer and returning to the depot – remains below 𝐷. Again, when routes are A. Corrêa et al.
open, the return distance to the depot is omitted from this calculation. Backhaul (B): Following prior work (Liu et al., 2024; Zhou et al., 2024a), we randomly designate 20% of customer nodes as backhaul customers – i.e., 10 nodes when 𝑛 = 50 and 20 nodes when 𝑛 = 100. During a route, the vehicle’s current capacity throughout a route must never exceed 𝐶 at any time. Consequently, during decoding, any backhaul customer whose 𝛿𝑖 would cause the vehicle to exceed its maximum capacity 𝐶 is masked out. Time window (TW): For time windows, we follow the procedure of Berto et al. (2025). If a vehicle arrives at a customer whose time window has not started yet, it must wait until 𝑒𝑖 to start service. Each depot also has an end time window, at which point all vehicles must have returned (when open routes are not present). During decoding, customers whose vehicles cannot complete service before the late time window 𝑙𝑖 are masked out. Inter-depot routes (I): Like open routes, inter-depot routes do not require extra data generation. A Boolean 𝑖𝑑𝑡 in the context vector indicates the presence of this constraint.
5.2. Main MDVRP results Our main results on the 24 MDVRP variants are shown in Table 4. Both FiLMMeD-MTPOMO and FiLMMeDMVMoE consistently outperform its respective neural baselines (MTPOMO and MVMoE) across the majority of datasets, obtaining lower average solution costs and performance gaps. On 50-node instances, FiLMMeD-MTPOMO reached an average gap of 4.717%, improving over its baseline by 2.888%, while FiLMMeD-MVMoE achieved 4.740%, outperforming MVMoE by 1.847%. On 100-node instances, FiLMMeD-MTPOMO and FiLMMeD-MVMoE averaged 6.847% and 6.433%, outperforming their counterparts by 1.921% and 5.288%, respectively. These results indicate the effectiveness of the proposed FiLM mechanism in the encoder, as well as the CL training strategy.
5.3. Results on single-depot variants (Zhou et al. (2024a) setting) The main results on single-depot VRPs are shown in Table 5. Overall FiLMMeD achieves competitive performance relative to its baselines. While no single model consistently prevails across all variants, FiLMMeD remains highly competitive in all settings. Specifically, on 50-node instances FiLMMeD-MTPOMO achieves an average gap of 4.138%, compared to 4.602% for MTPOMO, while FiLMMeDMVMoE reaches 4.081% versus 4.352% for MVMoE. On 100-node datasets, FiLMMeD-MTPOMO obtained an average gap of 4.099% (versus 4.969% from MTPOMO), and FiLMMeD-MVMoE 4.045% (against 4.595% from MVMoE). Importantly, we recall that while the baselines MTPOMO and MVMoE were trained for 100M instances (5000 epochs with 20,000 instances each), FiLMMeD was finetuned on 30M instances (300 epochs with 100,000 instances each). Even when accounting the pre-training phase on the Page 11 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
0.127% * 4.117% 4.002% 2.726% 3.507%
20.8m 3.5h 52s 49s 68s 69s
MDVRPB
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
7.348 7.347 7.618 7.607 7.588 7.596
0.017% * 3.688% 3.545% 3.280% 3.387%
10.4m 1.7h 12s 12s 17s 17s
10.645 10.627 11.203 11.248 11.225 11.186
0.169% * 5.436% 5.864% 5.647% 5.268%
20.8m 3.5h 52s 48s 64s 64s
MDVRPTW
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
12.527 12.269 12.609 12.550 12.595 12.565
1.901% * 2.749% 2.277% 2.631% 2.401%
10.4m 1.7h 17s 17s 23s 24s
21.540 19.722 20.610 20.577 22.020 20.533
8.746% * 4.485% 4.323% 11.629% 4.101%
20.8m 3.5h 75s 71s 97s 94s
MDOVRPB
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
5.194 5.194 5.774 5.456 5.605 5.450
* * 11.149% 5.031% 7.888% 4.924%
10.4m 1.7h 13s 14s 17s 19s
7.644 7.642 8.482 8.233 8.504 8.183
0.020% * 10.951% 7.706% 11.234% 7.053%
20.8m 3.5h 52s 52s 73s 70s
MDVRPBL
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
7.318 7.316 7.601 7.587 7.574 7.579
0.016% * 3.896% 3.709% 3.514% 3.593%
10.4m 1.7h 14s 14s 18s 18s
10.591 10.572 11.146 11.202 11.170 11.126
0.174% * 5.430% 5.956% 5.665% 5.239%
20.8m 3.5h 60s 54s 71s 70s
MDVRPLTW
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
12.492 12.167 12.491 12.443 12.494 12.461
2.473 * 2.640% 2.249% 2.664% 2.388%
10.4m 1.7h 18s 18s 25s 25s
21.083 19.614 20.498 20.479 21.933 20.441
7.192% * 4.489% 4.394% 11.801% 4.196%
20.8m 3.5h 85s 80s 106s 104s
MDVRPBLTW MDOVRPBTW
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.172 7.975 8.962 8.627 8.885 8.635
2.379% * 12.348% 8.147% 11.392% 8.244%
10.4m 1.7h 17s 19s 23s 25s
13.274 13.073 15.098 14.420 15.694 14.384
1.445% * 15.437% 10.269% 19.966% 9.991%
20.8m 3.5h 77s 76s 103s 97s
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
13.006 12.243 13.624 13.460 13.602 13.486
5.951% * 11.262% 9.935% 11.090% 10.151%
10.4m 1.7h 20s 19s 24s 24s
20.904 19.643 22.389 22.080 24.342 22.021
6.073% * 13.949% 12.393% 23.904% 12.097%
20.8m 3.5h 85s 78s 104s 99s
MDVRPI
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
7.930 7.876 8.178 8.173 8.141 8.168
0.677% * 3.866% 3.797% 3.387% 3.742%
10.4m 1.7h 14s 14s 19s 19s
11.895 11.643 12.161 12.236 12.111 12.181
2.159% * 4.482% 5.131% 4.048% 4.652%
20.8m 3.5h 55s 52s 73s 71s
MDVRPIL
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
7.972 7.918 8.214 8.211 8.185 8.203
0.681% * 3.758% 3.722% 3.391% 3.627%
10.4m 1.7h 14s 15s 20s 20s
11.938 11.676 12.205 12.274 12.159 12.215
2.234% * 4.561% 5.164% 4.161% 4.652%
20.8m 3.5h 62s 60s 81s 79s
MDVRPIBL
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
7.264 7.222 7.643 7.598 7.619 7.593
0.575% * 5.864% 5.245% 5.527% 5.166%
10.4m 1.7h 16s 14s 21s 18s
10.712 10.507 11.233 11.249 11.297 11.183
1.934% * 6.937% 7.091% 7.549% 6.463%
20.8m 3.5h 62s 56s 78s 73s
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
13.415 12.288 13.327 12.573 13.127 12.592
8.654% * 8.514% 2.297% 6.814% 2.451%
10.4m 1.7h 26s 19s 31s 25s
23.440 19.602 21.084 20.455 22.221 20.412
18.695% * 7.555% 4.337% 13.347% 4.115%
20.8m 3.5h 112s 83s 119s 107s
MDVRP, the total training represents 60% of the baselines training (30M instances from the pre-training phase plus 30M from the fine-tuning phase).
5.4. Ablation studies In this section, we assess the contribution of each component of our approach through a series of ablation studies. All models in this section were trained on instances with 50 nodes, and inference was performed using a greedy decoding with ×8 instance augmentation.
A. Corrêa et al.
Time
MDOVRP
11.823 11.808 12.294 12.280 12.129 12.222
𝑛 = 100 Gap
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
5.338 5.338 5.562 5.474 5.503 5.468
* * 4.166% 2.522% 3.071% 2.411%
10.4m 1.7h 13s 13s 18s 19s
7.960 7.958 8.404 8.375 8.390 8.334
0.026% * 5.573% 5.207% 5.401% 4.696%
20.8m 3.5h 52s 52s 69s 74s
MDVRPL
10.4m 1.7h 13s 13s 18s 18s
Obj.
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.008 8.007 8.225 8.184 8.222 8.171
0.009% * 2.708% 2.202% 2.669% 2.040%
10.4m 1.7h 14s 14s 20s 19s
11.820 11.803 12.288 12.280 12.126 12.220
0.140% * 4.103% 4.039% 2.735% 3.535%
20.8m 3.5h 61s 57s 76s 77s
MDOVRPTW
0.013% * 2.712% 2.163% 2.661% 1.980%
Time
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.114 7.981 8.421 8.092 8.369 8.104
1.653% * 5.498% 1.363% 4.852% 1.515%
10.4m 1.7h 17s 19s 24s 26s
13.477 13.143 14.212 13.581 14.401 13.542
2.407% * 8.117% 3.300% 9.510% 3.005%
20.8m 3.5h 75s 77s 98s 102s
MDOVRPL
7.995 7.994 8.212 8.167 8.207 8.152
𝑛 = 50 Gap
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
5.334 5.334 5.561 5.469 5.501 5.467
0.001% * 4.225% 2.501% 3.112% 2.473%
10.4m 1.7h 14s 15s 20s 21s
7.982 7.979 8.432 8.399 8.411 8.358
0.032% * 5.651% 5.243% 5.389% 4.724%
20.8m 3.5h 59s 60s 80s 81s
MDVRPBTW
MDVRP
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
Obj.
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
12.615 12.312 13.711 13.556 13.690 13.577
2.354% * 11.317% 10.067% 11.150% 10.246%
10.4m 1.7h 17s 17s 23s 23s
21.160 19.680 22.430 22.101 24.408 22.051
7.175% * 13.925% 12.280% 23.983% 12.026%
20.8m 3.5h 76s 70s 96s 90s
MDOVRPBL
Time
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
5.187 5.187 5.767 5.445 5.594 5.444
* * 11.169% 4.973% 7.840% 4.950%
10.4m 1.7h 13s 15s 18s 20s
7.649 7.647 8.485 8.236 8.506 8.192
0.026% * 10.926% 7.675% 11.192% 7.106%
20.8m 3.5h 58s 58s 77s 75s
MDOVRPLTW
𝑛 = 100 Gap
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.017 7.956 8.402 8.063 8.343 8.071
0.741% * 5.585% 1.321% 4.845% 1.419%
10.4m 1.7h 18s 20s 24s 27s
13.575 13.063 14.127 13.499 14.325 13.473
3.721% * 8.134% 3.317% 9.613% 3.114%
20.8m 3.5h 83s 85s 111s 111s
MDOVRPBLTW
Obj.
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.153 7.968 8.952 8.612 8.890 8.627
2.249% * 12.293% 8.052% 11.523% 8.230%
10.4m 1.7h 18s 20s 24s 26s
13.506 13.010 15.035 14.353 15.585 14.315
3.749% * 15.543% 10.312% 19.746% 10.020%
20.8m 3.5h 84s 83s 111s 105s
MDVRPIB
Time
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
7.234 7.191 7.634 7.592 7.607 7.579
0.598% * 6.195% 5.612% 5.816% 5.428%
10.4m 1.7h 15s 13s 19s 18s
10.655 10.456 11.174 11.193 11.243 11.127
1.900% * 6.897% 7.085% 7.566% 6.452%
20.8m 3.5h 56s 51s 71s 67s
MDVRPITW
𝑛 = 50 Gap
Method
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
13.129 12.285 13.347 12.570 13.100 12.584
6.410% * 8.740% 2.317% 6.622% 2.421%
10.4m 1.7h 24s 18s 29s 24s
24.548 19.597 21.086 20.452 22.197 20.411
24.425% * 7.598% 4.347% 13.260% 4.137%
20.8m 3.5h 97s 75s 110s 97s
MDVRPIBLTW MDVRPIBTW
Obj.
MDVRPILTW
Table 4 Testing results on 1K test instances of 24 MDVRPs. The best MTL results are shown in gray, and the models improved by our approach are underlined.
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
13.174 12.252 14.614 13.486 14.251 13.514
7.318% * 19.416% 10.091% 16.335% 10.324%
10.4m 1.7h 25s 18s 28s 23s
24.149 19.516 23.057 21.953 24.524 21.894
23.231% * 18.135% 12.470% 25.622% 12.168%
20.8m 3.5h 93s 72s 108s 93s
HGS-PyVRP HGS-PyVRP (10x) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
13.297 12.213 14.495 13.447 14.172 13.467
8.480% * 18.765% 10.068% 16.005% 10.237%
10.4m 1.7h 26s 19s 30s 24s
23.194 19.603 23.133 22.042 24.627 21.973
17.481% * 17.996% 12.421% 25.620% 12.076%
20.8m 3.5h 105s 80s 116s 102s
Method
5.4.1. Curriculum learning To quantify the effectiveness of the CL regimen, we trained FiLMMeD-MTPOMO and FiLMMeD-MVMoE under three alternative strategies: 1) unified, which consists in training on variants that contain only a single constraint (i.e., MDVRP, MDOVRP, MDVRPB, MDVRPL, MDVRPTW, MDVRPI), following Liu et al. (2024); 2) full task set, which consists in sampling uniformly across all available variants, following Berto et al. (2025); 3) standard CL, in which the initial training problem set contains only singleconstraint variants. In Section 4.3.1, we argue that including three additional variants in the initial CL training set (i.e.,
Page 12 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
* 0.115% 0.788% 0.806% 0.987% 1.185% 0.896% 1.082%
4.6m 9.9m 1.7h 3s 3s 4s 4s 5s
15.504 15.590 15.935 15.734 15.790 15.797 15.760 15.789
* 0.556% 2.751% 1.488% 1.846% 1.895% 1.653% 1.841%
9.1m 18.0m 3.5h 9s 9s 15s 11s 20s
OVRP
LKH3 OR-Tools (x10) POMO MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
6.511 6.498 6.609 6.671 6.665 6.655 6.662
0.198% * 1.685% 2.634% 2.547% 2.402% 2.505%
4.5m 1.7h 2s 2s 4s 3s 5s
9.828 9.842 10.044 10.169 10.113 10.138 10.111
* 0.122% 2.192% 3.458% 2.888% 3.136% 2.865%
5.3m 3.5h 8s 8s 15s 10s 21s
VRPB
OR-Tools (x10) POMO MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.046 8.149 8.182 8.201 8.170 8.190
* 1.276% 1.684% 1.919% 1.540% 1.789%
1.7h 2s 2s 3s 3s 4s
11.878 11.993 12.072 12.068 12.027 12.059
* 0.995% 1.674% 1.636% 1.285% 1.554%
3.5h 7s 7s 13s 9s 17s
OVRPB
OR-Tools (x10) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
5.745 6.116 6.000 6.092 5.999
* 6.430% 4.397% 5.999% 4.380%
1.7h 2s 4s 3s 5s
8.365 8.979 8.756 8.959 8.759
* 7.335% 4.667% 7.088% 4.703%
3.5h 8s 14s 9s 18s
VRPBL
OR-Tools (x10) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.029 8.188 8.194 8.172 8.185
* 1.971% 2.057% 1.776% 1.941%
1.7h 2s 4s 3s 5s
11.790 11.998 11.986 11.945 11.975
* 1.793% 1.702% 1.346% 1.603%
3.5h 8s 15s 9s 19s
VRPLTW
OR-Tools (x10) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
14.598 14.961 14.999 14.937 14.978
* 2.586% 2.835% 2.421% 2.696%
1.7h 3s 5s 4s 6s
25.195 25.619 25.628 25.514 25.610
* 1.920% 1.942% 1.471% 1.854%
3.5h 12s 22s 13s 28s
OVRPBTW
OR-Tools (x10) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.675 9.514 9.415 9.486 9.390
* 9.628% 8.504% 9.308% 8.216%
1.7h 3s 4s 4s 6s
14.384 15.879 15.619 15.808 15.607
* 10.453% 8.653% 9.948% 8.572%
3.5h 10s 17s 11s 22s
OR-Tools (x10) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
14.677 15.980 15.834 15.945 15.794
* 9.035% 8.033% 8.775% 7.768%
1.7h 3s 5s 4s 6s
25.342 27.247 26.818 27.142 26.801
* 7.746% 6.053% 7.332% 5.985%
3.5h 11s 20s 12s 25s
MDOVRPTW, MDVRPBTW and MDVRPITW) is beneficial to generalization. Hence, we include the standard CL results as well, which does not feature these three variants in the initial training set. Table 6 Average objective function and gaps across 24 MDVRP variants for different training strategies. Model
Obj.
Gap
FiLMMeD-MTPOMO - unified FiLMMeD-MTPOMO - full task set FiLMMeD-MTPOMO - standard CL FiLMMeD-MTPOMO
9.630 9.320 9.306 9.268
8.396% 5.295% 5.134% 4.717%
FiLMMeD-MVMoE - unified FiLMMeD-MVMoE - full task set FiLMMeD-MVMoE - standard CL FiLMMeD-MVMoE
9.730 9.298 9.299 9.273
9.318% 5.046% 5.020% 4.740%
All three baselines also include the FiLM mechanism. Table 6 reports the average objective and gap for each model under the different training strategies. We observed that A. Corrêa et al.
Time
Obj.
𝑛 = 100 Gap
Time
VRPTW
10.334 10.346 10.418 10.418 10.437 10.457 10.428 10.446
𝑛 = 50 Gap
HGS LKH3 OR-Tools (x10) POMO MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
14.509 14.607 14.665 14.940 15.032 15.064 14.999 15.046
* 0.664% 1.011% 2.990% 3.637% 3.869% 3.410% 3.736%
8.4m 5.5m 1.7h 3s 3s 4s 4s 6s
24.339 24.721 25.212 25.367 25.610 25.619 25.512 25.605
* 1.584% 3.482% 4.307% 5.313% 5.327% 4.903% 5.268%
19.6m 7.8m 3.5h 11s 11s 20s 12s 26s
VRPL
CVRP
HGS LKH3 OR-Tools (x10) POMO MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
Obj.
LKH3 OR-Tools (x10) POMO MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
10.571 10.495 10.491 10.513 10.527 10.501 10.516
0.790% * -0.008% 0.201% 0.333% 0.092% 0.233%
7.8m 1.7h 2s 2s 4s 3s 5s
15.771 16.004 15.785 15.846 15.849 15.812 15.845
* 1.444% 0.093% 0.479% 0.502% 0.261% 0.471%
16.0m 3.5h 9s 9s 17s 10s 22s
OVRPTW
Time
OR-Tools (x10) POMO MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.683 8.891 8.987 8.989 8.964 8.973
* 2.377% 3.470% 3.499% 3.210% 3.312%
1.7h 3s 3s 4s 4s 6s
14.380 14.728 15.008 14.956 14.927 14.939
* 2.467% 4.411% 4.048% 3.852% 3.929%
3.5h 10s 10s 18s 11s 24s
OVRPL
𝑛 = 100 Gap
OR-Tools (x10) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
6.490 6.668 6.660 6.650 6.659
* 2.734% 2.615% 2.454% 2.582%
1.7h 2s 4s 3s 6s
9.790 10.126 10.070 10.097 10.070
* 3.441% 2.868% 3.148% 2.869%
3.5h 9s 17s 10s 23s
VRPBTW
Obj.
OR-Tools (x10) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
14.771 16.055 15.886 16.022 15.858
* 8.841% 7.708% 8.600% 7.533%
1.7h 3s 4s 4s 6s
25.496 27.319 26.912 27.236 26.887
* 7.413% 5.822% 7.078% 5.719%
3.5h 10s 18s 11s 23s
OVRPBL
Time
OR-Tools (x10) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
5.739 6.104 5.991 6.076 5.986
* 6.306% 4.364% 5.843% 4.260%
1.7h 2s 4s 3s 5s
8.348 8.961 8.746 8.942 8.740
* 7.343% 4.768% 7.115% 4.705%
3.5h 8s 15s 9s 19s
OVRPLTW
𝑛 = 50 Gap
Method
OR-Tools (x10) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.669 8.987 8.990 8.966 8.971
* 3.633% 3.667% 3.396% 3.440%
1.7h 3s 5s 4s 6s
14.279 14.896 14.854 14.828 14.854
* 4.374% 4.080% 3.903% 4.083%
3.5h 11s 20s 12s 26s
OVRPBLTW
Obj.
VRPBLTW
Table 5 Fine-tuning performance on 1K test instances of 16 single-depot VRPs, following the setting of Zhou et al. (2024a). The best MTL results are shown in gray, and the models improved by our approach are underlined.
OR-Tools (x10) MTPOMO FiLMMeD-MTPOMO MVMoE/4E FiLMMeD-MVMoE/4E
8.673 9.532 9.430 9.503 9.415
* 9.851% 8.679% 9.516% 8.505%
1.7h 3s 5s 4s 6s
14.250 15.738 15.486 15.671 15.481
* 10.498% 8.732% 10.009% 8.692%
3.5h 10s 18s 11s 23s
Method
our CL training yields significant generalization improvements over the other three strategies on both FiLMMeDMTPOMO and FiLMMeD-MVMoE. This confirms that, for MDVRP variants, a structured curriculum is essential for robust generalization. Simpler strategies like unified sampling – which relies on zero-shot composition – struggled on the multi-depot setting. On variants with two, three and four constraints simultaneously, it struggled to learn how these different constraints interact with one another. In contrast to the single-depot VRP, learning constraints in isolation proved insufficient for the MDVRP. One possible reason for this stems from the fact that the multi-depot structure itself acts as an additional constraint, complicating constraint interactions significantly more than in singledepot VRPs. The standard CL strategy improves upon this problem by gradually increasing the complexity of variants sampled throughout training. However, the delayed exposure to complex variants led to stagnated learning. In contrast, our CL regimen, which exposes the model to more complex constraint interactions early on, led to a smoother convergence throughout training. We also conducted sensitivity analyses on different CL schedules, i.e., when to add more complex variants into
Page 13 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
the training problem set. We examined two alternative curriculum schedules: 1) a faster progression (transitions at 25%, 50%, and 75% of epochs); and 2) a slower progression (transitions at 50%, 70%, and 90% of epochs). Figure 2 compares the learning curves of these alternatives against our selected schedule (which has transitions at 30%, 60%, and 90% of epochs). Here, we present the results from epoch 50 onward to improve visualization. Overall, our chosen schedule led to a much better generalization compared to the other two alternatives.
Figure 3: Convergence of FiLMMeD models during fine-tuning with PO and RL (smoothed over a moving average of 20 epochs for better visualization).
Figure 4: Average gap on 16 single-depot VRP variants of fine-tuned FiLMMeD models with PO and RL.
support its adoption as a superior alternative to RL in future MTL models. Figure 2: Comparison of different CL schedules: Evolution of the average testing objective cost across 24 MDVRP variants.
Table 7 Average gaps across all 24 MDVRP variants: RL vs. PO comparison Model
5.4.2. Preference optimization To assess the impact of fine-tuning FiLMMeD on singledepot variants via PO, we also fine-tuned FiLMMeD for 300 additional epochs with the standard REINFORCE with shared baselines loss, keeping all configurations and hyperparameters identical. Figure 3 reports the convergence of both approaches over these epochs. Both FiLMMeDMTPOMO and FiLMMeD-MVMoE display a more favorable convergence profile when fine-tuned with PO rather than RL. We further compare the two strategies across the 16 single-depot VRP variants, reporting their average performance gaps during inference in Figure 4. The results validate the convergence improvements observed during fine-tuning, with both models outperforming their respective RL variants. As stated previously, we decided to pre-train MDVRP models with RL to better isolate the effects of the CL regimen and FiLM. However, for the sake of comparison, we also assessed the performance of PO when pretraining the MDVRP models. Table 7 shows that PO works just as well in the multi-depot setting, yielding consistent improvements over RL across both FiLMMeD-MTPOMO and FiLMMeD-MVMoE architectures. These results further
A. Corrêa et al.
FiLMMeD-MTPOMO FiLMMeD-MVMoE
RL
PO
4.717% 4.740%
4.439% 4.509%
5.4.3. FiLM Lastly, we evaluate the contribution of the FiLM mechanism. For this analysis, we fine-tuned MTPOMO and MVMoE, starting from the pre-trained models released by Zhou et al. (2024a)1 . Both models were originally trained on single-depot variants, and our goal was to extend their capabilities to also handle MDVRP variants. To extend their capabilities to multi-depot settings while testing compositional generalization, we fine-tuned both architectures with and without FiLM on a subset of 12 variants: 6 single-depot (CVRP, OVRP, VRPTW, VRPL, VRPB, OVRPTW) and 6 multi-depot (MDVRP, MDVRPB, MDVRPL, MDOVRP, MDVRPTW, MDVRPI), holding out the remaining 28 variants for zero-shot evaluation. We incorporated FiLM by extending the conditioning vector 𝑧 to 6 dimensions, adding a Boolean feature indicating whether an instance contains one or multiple depots. To preserve the prior knowledge from the pre-trained models, 1 https://github.com/RoyalSkye/Routing-MVMoE/tree/main/pretrained
Page 14 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
Figure 5: Convergence of MTPOMO and MVMoE fine-tuned with and without FiLM on both single- and multi-depot variants.
we initialized the FiLM parameters as follows: 𝛾 weights set to zero with bias set to one, and 𝛽 weights and bias set to zero. This ensures that, at initialization, the FiLM layers behave as identity transformations and do not alter the existing representations. We then fine-tuned both the FiLMaugmented and original (non-FiLM) models for 500 epochs (approximately 10% of the original total training data), using 20,000 training instances per epoch. To isolate the architectural contribution of FiLM and ensure fair comparison with the original pre-trained models, both the FiLM-augmented and non-FiLM models were trained using the REINFORCE with shared baselines algorithm. The convergence across the 500 fine-tuning epochs is shown in Figure 5. Overall, the FiLM component resulted in a clearly smoother convergence on both models. This translated into substantially better inference performance, particularly on zero-shot generalization to unseen constraint combinations. As reported in Table 8, FiLMaugmented models consistently outperformed their nonFiLM counterparts across both seen and unseen variants. FiLM-augmented MTPOMO achieved an average gap of 5.437% compared to 5.972% for the baseline MTPOMO, with the most pronounced gains observed on the 28 held-out unseen variants (6.764% vs. 7.386%). Similarly, FiLMMeDaugmented MVMoE attained an average gap of 5.297% versus 5.716% for the standard MVMoE, with unseen variant performance improving from 7.098% to 6.620%. These results confirm our prior intuition that FiLM would enhance the performance on unseen constraint combinations. Table 8 Generalization performance on seen vs. unseen variants. Models trained on 12 variants and evaluated zero-shot on 28 heldout variants. Methods
Seen (12)
Unseen (28)
Average
MTPOMO MTPOMO + FiLM
2.672% 2.341%
7.386% 6.764%
5.972% 5.437%
MVMoE MVMoE + FiLM
2.493% 2.208%
7.098% 6.620%
5.716% 5.297%
A. Corrêa et al.
Figure 6: T-SNE visualization comparison for the learned customer embeddings before and after the FiLM mechanism.
Figure 7: T-SNE visualization comparison for the last encoder layer of different models.
To provide more empirical evidence regarding the effectiveness of the FiLM mechanism, we analyzed the latent representations of all 24 MDVRP variants using the t-SNE technique (van der Maaten and Hinton, 2008). First, we compared the learned customer embeddings directly before and after the FiLM transformation. The pre-modulation embeddings (on the left of Figure 6) reveal a latent space structured primarily by input data topology. We can see that two large cluster emerge, distinguishing variants with and without time windows. In this case, variants that differ by other constraints (such as open routes and backhauls) remain indistinguishable and entangled within these clusters. This indicates that the linear layer (the same from equation 2) learns a generally task-agnostic unified representation, capturing fundamental spatial and demand features shared across all variants. After applying FiLM, the latent space undergoes a severe reorganization (shown on the right of Figure 6). The embeddings are now organized into very distinct and separated clusters corresponding to each constraint combination, suggesting that the affine transformations performed by FiLM Page 15 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP Table 9 Performance on 1K test instances of 16 single-depot VRPs, following the setting of Berto et al. (2025). The best MTL results (among neural models) are shown in gray, and the models improved by our approach (FiLMMeD vs. respective CaDA) are underlined. 𝑛 = 100
* 1.907% 1.423% 1.229% 1.257% 1.182% 0.924% 1.012% 0.865%
10.4m 10.4m 2s 3s 2s 2s 4s 5s 5s
15.628 16.280 15.941 15.888 15.860 15.870 15.793 15.800 15.762
* 4.178% 2.030% 1.693% 1.524% 1.578% 1.089% 1.139% 0.894%
20.8m 20.8m 8s 11s 8s 8s 19s 28s 28s
OVRP
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
6.507 6.553 6.717 6.705 6.682 6.670 6.649 6.658 6.647
* 0.686% 3.194% 3.003% 2.658% 2.468% 2.148% 2.293% 2.128%
10.4m 10.4m 2s 3s 2s 2s 4s 5s 5s
9.725 9.995 10.216 10.177 10.115 10.121 10.078 10.078 10.030
* 2.732% 5.028% 4.617% 3.996% 4.045% 3.605% 3.616% 3.115%
20.8m 20.8m 8s 11s 8s 8s 19s 29s 29s
VRPB
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
9.687 9.802 10.036 10.007 9.979 9.960 9.925 9.931 9.909
* 1.159% 3.596% 3.292% 3.000% 2.800% 2.442% 2.502% 2.278%
10.4m 10.4m 2s 3s 2s 2s 4s 5s 5s
14.377 14.933 15.102 15.023 14.935 14.960 14.859 14.852 14.801
* 3.853% 5.052% 4.505% 3.906% 4.038% 3.369% 3.319% 2.964%
20.8m 20.8m 8s 10s 8s 8s 19s 28s 28s
VRPBL
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
10.186 10.331 10.679 10.639 10.569 10.543 10.512 10.511 10.497
* 1.390% 4.760% 4.384% 3.713% 3.461% 3.152% 3.143% 3.005%
10.4m 10.4m 2s 3s 2s 2s 4s 5s 5s
14.779 15.426 15.718 15.642 15.523 15.525 15.424 15.397 15.342
* 4.338% 6.294% 5.771% 5.008% 5.001% 4.318% 4.153% 3.776%
20.8m 20.8m 8s 11s 8s 8s 19s 29s 29s
VRPBTW
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
18.292 18.366 18.649 18.632 18.573 18.500 18.497 18.524 18.484
* 0.383% 1.938% 1.841% 1.517% 1.117% 1.105% 1.251% 1.034%
10.4m 10.4m 2s 3s 2s 2s 4s 6s 6s
29.467 29.945 30.478 30.437 30.249 30.059 30.042 30.115 30.076
* 1.597% 3.426% 3.284% 2.641% 1.999% 1.938% 2.188% 2.059%
20.8m 20.8m 9s 12s 9s 9s 21s 32s 31s
OVRPB
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
6.898 6.928 7.105 7.089 7.065 7.049 7.027 7.038 7.024
* 0.412% 2.973% 2.744% 2.385% 2.159% 1.844% 2.003% 1.802%
10.4m 10.4m 2s 3s 2s 2s 4s 6s 6s
10.335 10.577 10.882 10.841 10.774 10.762 10.695 10.708 10.655
* 2.315% 5.264% 4.869% 4.233% 4.099% 3.456% 3.592% 3.071%
20.8m 20.8m 8s 11s 8s 8s 20s 30s 29s
OVRPBLTW
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
11.668 11.681 11.823 11.816 11.789 11.760 11.754 11.773 11.747
* 0.106% 1.315% 1.249% 1.017% 0.771% 0.721% 0.885% 0.665%
10.4m 10.4m 3s 4s 2s 2s 4s 7s 6s
19.156 19.305 19.658 19.640 19.554 19.435 19.437 19.479 19.456
* 0.767% 2.602% 2.514% 2.061% 1.439% 1.441% 1.666% 1.546%
20.8m 20.8m 9s 12s 9s 9s 22s 33s 34s
OVRPL
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
6.507 6.552 6.720 6.706 6.683 6.671 6.650 6.658 6.647
* 0.668% 3.248% 3.028% 2.680% 2.475% 2.157% 2.295% 2.122%
10.4m 10.4m 2s 3s 2s 2s 4s 5s 5s
9.724 10.001 10.224 10.184 10.121 10.122 10.078 10.081 10.027
* 2.791% 5.112% 4.693% 4.054% 4.052% 3.604% 3.647% 3.089%
20.8m 20.8m 8s 11s 8s 8s 20s 29s 29s
successfully disentangle task specific features from general ones. After, we compared the learned representations of both FiLMMeD models and its baselines by extracting the learned node embeddings from the last encoder layer, that is, ℎ(𝐿) 𝑖 , ∀𝑣𝑖 ∈ . Figure 7 presents the t-SNE projections of these embeddings for FiLMMeD-MTPOMO and
Time
Obj.
Gap
Time
VRPTW
10.372 10.572 10.520 10.499 10.502 10.494 10.468 10.476 10.461
Gap
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
16.031 16.089 16.419 16.400 16.341 16.278 16.266 16.291 16.251
* 0.347% 2.423% 2.298% 1.933% 1.536% 1.461% 1.616% 1.365%
10.4m 10.4m 2s 3s 2s 2s 4s 6s 6s
25.423 25.814 26.433 26.390 26.228 26.070 26.029 26.106 26.062
* 1.506% 3.962% 3.789% 3.154% 2.530% 2.365% 2.673% 2.494%
20.8m 20.8m 9s 11s 8s 8s 20s 31s 31s
VRPL
CVRP
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
𝑛 = 100
Obj.
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
10.587 10.570 10.775 10.753 10.747 10.731 10.706 10.711 10.699
* 2.343% 1.733% 1.525% 1.485% 1.333% 1.101% 1.152% 1.044%
10.4m 10.4m 2s 3s 2s 2s 4s 5s 5s
15.766 16.466 16.157 16.099 16.057 16.057 15.988 15.987 15.950
* 5.302% 2.483% 2.113% 1.858% 1.847% 1.419% 1.424% 1.181%
20.8m 20.8m 8s 11s 8s 8s 19s 29s 29s
OVRPTW
Time
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
10.510 10.519 10.676 10.674 10.645 10.613 10.611 10.623 10.597
* 0.078% 1.558% 1.541% 1.264% 0.957% 0.940% 1.058% 0.810%
10.4m 10.4m 2s 3s 2s 2s 4s 6s 6s
16.926 17.027 17.442 17.416 17.328 17.226 17.214 17.247 17.232
* 0.583% 3.022% 2.870% 2.352% 1.751% 1.678% 1.876% 1.781%
20.8m 20.8m 9s 12s 9s 9s 19s 32s 33s
VRPBLTW
Gap
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
18.361 18.422 19.001 18.983 18.910 18.848 18.848 18.869 18.832
* 0.332% 2.199% 2.097% 1.713% 1.376% 1.369% 1.484% 1.288%
10.4m 10.4m 3s 3s 2s 2s 4s 5s 6s
29.026 29.830 30.948 30.892 30.705 30.520 30.499 30.556 30.523
* 2.770% 3.794% 3.609% 2.978% 2.359% 2.287% 2.483% 2.371%
20.8m 20.8m 9s 12s 9s 9s 19s 32s 32s
VRPLTW
Obj.
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
16.356 16.441 16.832 16.817 16.728 16.669 16.661 16.684 16.636
* 0.499% 2.877% 2.783% 2.248% 1.879% 1.836% 1.984% 1.688%
10.4m 10.4m 2s 3s 2s 2s 4s 6s 6s
25.757 26.259 26.913 26.866 26.706 26.540 26.498 26.548 26.510
* 1.899% 4.455% 4.272% 3.645% 2.995% 2.836% 3.043% 2.883%
20.8m 20.8m 9s 12s 9s 9s 20s 31s 31s
OVRPBL
Time
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
6.899 6.927 7.112 7.094 7.068 7.051 7.028 7.040 7.025
* 0.386% 3.053% 2.799% 2.417% 2.166% 1.846% 2.011% 1.797%
10.4m 10.4m 2s 3s 2s 2s 4s 5s 5s
10.335 10.582 10.888 10.847 10.778 10.762 10.695 10.709 10.655
* 2.363% 5.318% 4.929% 4.266% 4.102% 3.459% 3.598% 3.074%
20.8m 20.8m 8s 11s 8s 8s 20s 29s 30s
OVRPBTW
Gap
A. Corrêa et al.
𝑛 = 50
Method
Obj.
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
11.669 11.682 11.823 11.816 11.790 11.761 11.754 11.774 11.747
* 0.109% 1.307% 1.245% 1.027% 0.779% 0.719% 0.892% 0.655%
10.4m 10.4m 3s 4s 2s 2s 4s 6s 7s
19.156 19.303 19.656 19.637 19.555 19.436 19.435 19.478 19.454
* 0.757% 2.592% 2.499% 2.062% 1.441% 1.434% 1.663% 1.537%
20.8m 20.8m 9s 13s 9s 9s 22s 34s 33s
OVRPLTW
𝑛 = 50
Method
HGS-PyVRP OR-Tools MTPOMO MVMoE RouteFinder CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
10.510 10.497 10.677 10.677 10.646 10.613 10.609 10.623 10.598
* 0.114% 1.572% 1.564% 1.267% 0.961% 0.926% 1.060% 0.819%
10.4m 10.4m 2s 3s 2s 2s 4s 6s 6s
16.926 17.023 17.442 17.418 17.328 17.226 17.214 17.245 17.232
* 0.728% 3.020% 2.880% 2.352% 1.752% 1.678% 1.870% 1.785%
20.8m 20.8m 9s 12s 9s 9s 21s 32s 32s
FiLMMeD-MVMoE and their respective baselines. Overall, we can see that FiLMMeD exhibits more structured and well-defined clusters, particularly for variants involving route length limits, backhauls, and time windows. This suggests that the FiLM mechanism effectively helps the model modulate its internal representations, allowing it to better adapt towards diverse constraint combinations.
Page 16 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP Table 10 Testing results on Cordeau MDVRP benchmark instances. The best neural-based method results are shown in gray, and the models improved by FiLMMeD are underlined.
Variant
HGS-PyVRP Obj. Gap Time
Obj.
MTPOMO Gap
Time
FiLMMeD-MTPOMO Obj. Gap Time
Obj.
MVMoE/4E Gap
Time
FiLMMeD-MVMoE/4E Obj. Gap Time
MDVRP MDOVRP MDVRPB MDVRPL MDVRPTW MDOVRPTW MDOVRPB MDOVRPL MDVRPBL MDVRPBTW MDVRPLTW MDOVRPBL MDOVRPBTW MDOVRPLTW MDVRPBLTW MDOVRPBLTW MDVRPI MDVRPIB MDVRPIL MDVRPITW MDVRPIBL MDVRPIBTW MDVRPILTW MDVRPIBLTW
1851.19 1386.02 1756.56 1898.82 3098.46 2153.43 1374.08 1385.97 1760.30 3120.24 3086.58 1373.20 2164.82 2167.34 3123.13 2156.32 1870.61 1771.57 1873.33 3084.07 1776.03 3097.93 3071.28 3069.37
* * * * * * * * * * * * * * * * * * * * * * * *
5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m 5m
2034.98 1722.21 1964.27 2035.81 3476.98 2761.57 1782.25 1722.21 1965.26 3715.97 3454.22 1798.56 2917.39 2775.88 3735.69 2901.50 2048.13 2009.43 2052.86 4437.59 2028.43 4797.86 4496.17 4848.59
9.031% 22.915% 10.999% 6.704% 11.196% 25.751% 28.145% 22.918% 10.469% 18.304% 10.749% 29.549% 32.298% 25.798% 18.527% 31.718% 9.479% 12.889% 9.624% 44.428% 13.655% 55.065% 44.037% 59.088%
7s 7s 8s 9s 10s 10s 8s 8s 9s 11s 11s 8s 11s 11s 12s 12s 8s 9s 10s 14s 10s 15s 16s 16s
2046.08 1541.42 1984.85 2040.60 3414.85 2394.81 1573.51 1540.99 1990.50 3644.22 3394.27 1575.64 2497.26 2412.36 3677.44 2520.55 2073.85 2026.46 2073.89 3394.66 2011.70 3647.92 3415.15 3578.70
10.062% 9.852% 12.472% 7.327% 9.426% 9.788% 13.156% 10.177% 12.380% 15.843% 8.823% 13.698% 14.340% 10.179% 17.156% 14.978% 10.965% 14.129% 10.805% 8.963% 13.287% 16.986% 10.023% 15.900%
7s 8s 8s 8s 10s 11s 8s 9s 9s 11s 12s 9s 12s 12s 12s 13s 9s 9s 10s 11s 10s 11s 12s 12s
1993.30 1762.79 1940.64 1993.35 3515.88 2789.77 1809.60 1762.79 1953.89 3873.42 3490.28 1813.13 2979.77 2785.94 3820.79 2928.39 2040.77 2033.86 2033.39 3609.96 2035.19 3931.74 3589.49 3869.80
6.951% 26.103% 9.465% 4.629% 12.521% 27.329% 30.541% 26.106% 10.018% 23.343% 11.739% 31.042% 36.011% 26.917% 21.443% 33.111% 8.963% 13.889% 8.586% 16.672% 14.109% 26.356% 16.553% 26.123%
9s 9s 10s 11s 13s 13s 10s 10s 10s 14s 14s 10s 13s 14s 15s 14s 11s 11s 12s 14s 12s 14s 15s 16s
2044.86 1538.02 1974.97 2037.33 3389.29 2342.40 1559.18 1544.24 1980.99 3632.57 3370.09 1570.14 2479.57 2378.96 3638.68 2493.31 2075.04 2023.05 2068.03 3359.50 2004.44 3588.74 3369.62 3550.08
9.969% 9.874% 11.452% 7.180% 8.600% 7.757% 12.140% 10.244% 11.644% 15.488% 8.230% 13.322% 13.378% 8.680% 15.225% 13.845% 11.000% 14.125% 10.385% 7.986% 13.279% 15.070% 8.674% 14.561%
9s 10s 10s 11s 12s 13s 10s 11s 10s 13s 14s 11s 14s 15s 14s 15s 11s 11s 12s 13s 12s 14s 15s 15s
Avg.
2227.94
*
5m
2811.83
23.472%
10s
2519.65
12.113%
10s
2681.58
19.522%
12s
2500.55
11.338%
12s
5.5. Results on single-depot variants (Berto et al. (2025) setting) Like Liu et al. (2025), we evaluate our approach across two distinct standardized experimental settings from the literature. This section details the results for 16 singledepot VRP variants using the setting defined by Berto et al. (2025). For these experiments, we trained a variation of CaDA (Li et al., 2025) from scratch for 300 epochs. This version, denoted as FiLMMeD-CaDA, incorporates the key contributions proposed in this work while retaining the core dual-attention architecture of CaDA. Specifically, we added the proposed FiLM mechanism to the encoder. Furthermore, instead of the traditional REINFORCE loss, we employed PO for training, setting 𝛼 = 0.03, following the recommendations of Pan et al. (2025a). Since our curriculum was designed for the MDVRP, we maintained the same mixed batch training strategy from the original work (where a single batch can contain instances from multiple different variants). All the key hyper-parameters of CaDA were kept the same for FiLMMeD-CaDA. We also include an additional FiLMMeD-CaDA† variation, which combines FiLMMeD-CaDA with ReLD (Huang et al., 2025). For neural baselines, we include results on MTPOMO, MVMoE, RouteFinder (in particular, their modern Transformer-based Encoder version), CaDA and CaDA†. Table 9 presents the main results on 16 single-depot VRP variants, of 50 and 100 nodes. Overall, both FiLMMeDCaDA and FiLMMeD-CaDA† consistently outperformed their non-FiLMMeD counterparts across the majority of A. Corrêa et al.
cases, further consolidating the effectiveness of our proposed contributions. Additionally, FiLMMeD-CaDA† displayed the best overall results across all other baselines, with an average performance gap of 1.46% on 50-node instances, and 2.35% on 100-node instances.
5.6. Results on benchmark datasets We also evaluate the performance of previous models on benchmark datasets. For the MDVRP, we generated new datasets derived from the 10 instances of Cordeau et al. (1997). While often cited as MDVRP problems, these instances originally imposed route length limits (MDVRPL). We utilized their original node coordinates and demands to create 10 corresponding instances for each of the 23 remaining variants, totaling 240 benchmark instances. Missing attributes were generated following Section 5.1: 20% of customers were randomly assigned as backhauls, and time windows were generated using the standard protocol. All models were evaluated using a greedy decoding with ×8 instance augmentation. For each instance, we ran HGSPyVRP with a 5-minute runtime limit. Table 10 presents the results, averaged across the 10 instances for each variant. Once again, both FiLMMeD models showed a significantly better performance than their respective baselines across most variants. For single-depot VRPs, we utilized the models from Table 9, that is, FiLMMeD-CaDA and FiLMMeD-CaDA†. Here, we considered the Set-X benchmark (Uchoa et al., 2017) from CVRPLib, which contains instances ranging from 100 to 1000 nodes. Similar to the previous MDVRP Page 17 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP Table 11 Results on CVRPLib instances from the Set-X (Uchoa et al., 2017). The best individual results are shown in gray. Set-X
MTPOMO
MVMoE
RouteFinder
CaDA
FiLMMeD-CaDA
CaDA†
FiLMMeD-CaDA†
DBDA
Instance
Opt.
Cost
Gap
Cost
Gap
Cost
Gap
Cost
Gap
Cost
Gap
Cost
Gap
Cost
Gap
Cost
Gap
X-n101-k25 X-n106-k14 X-n110-k13 X-n115-k10 X-n120-k6 X-n125-k30 X-n129-k18 X-n134-k13 X-n139-k10 X-n143-k7 X-n148-k46 X-n153-k22 X-n157-k13 X-n162-k11 X-n167-k10 X-n172-k51 X-n176-k26 X-n181-k23 X-n186-k15 X-n190-k8 X-n195-k51 X-n200-k36 X-n204-k19 X-n209-k16 X-n214-k11 X-n219-k73 X-n223-k34 X-n228-k23 X-n233-k16 X-n237-k14 X-n242-k48 X-n247-k50
27591 26362 14971 12747 13332 55539 28940 10916 13590 15700 43448 21220 16876 14138 20557 45607 47812 25569 24145 16980 44225 58578 19565 30656 10856 117595 40437 25742 19230 27042 82751 37274
29470 28029 15100 13433 14051 59015 30176 11707 14058 16626 46648 23514 17922 14616 21662 48960 51989 26572 25236 18369 48310 62041 20652 32333 11699 121980 43381 28523 20644 30047 88179 41610
6.810% 6.323% 0.862% 5.382% 5.393% 6.259% 4.271% 7.246% 3.444% 5.898% 7.365% 10.811% 6.198% 3.381% 5.375% 7.352% 8.736% 3.923% 4.519% 8.180% 9.237% 5.912% 5.556% 5.470% 7.765% 3.729% 7.280% 10.803% 7.353% 11.112% 6.559% 11.633%
29076 27443 15327 13475 13782 58200 29334 11462 14099 16349 45893 23661 17439 14705 21504 47883 52117 26456 25151 19078 46974 61627 20584 32358 11597 124434 42694 28033 20656 29772 87497 40973
5.382% 4.101% 2.378% 5.711% 3.375% 4.791% 1.361% 5.002% 3.745% 4.134% 5.627% 11.503% 3.336% 4.010% 4.607% 4.990% 9.004% 3.469% 4.166% 12.356% 6.216% 5.205% 5.208% 5.552% 6.826% 5.816% 5.582% 8.900% 7.415% 10.095% 5.735% 9.924%
29048 27159 15314 13338 13765 58570 29457 11624 13812 16257 45026 23478 17315 14664 21425 48162 51501 26097 25153 17871 47396 61139 20531 31876 11668 120344 42251 28699 20761 29595 85704 40642
5.281% 3.023% 2.291% 4.636% 3.248% 5.457% 1.786% 6.486% 1.634% 3.548% 3.632% 10.641% 2.601% 3.720% 4.222% 5.602% 7.716% 2.065% 4.175% 5.247% 7.170% 4.372% 4.937% 3.980% 7.480% 2.338% 4.486% 11.487% 7.962% 9.441% 3.569% 9.036%
28944 27042 15229 13060 13678 57748 29500 11652 13940 16189 45606 23142 17295 14704 21078 48198 51120 26262 25345 17882 46723 61010 20735 32184 11748 120011 42273 27821 20285 30282 85813 39918
4.904% 2.579% 1.723% 2.455% 2.595% 3.977% 1.935% 6.742% 2.575% 3.115% 4.967% 9.057% 2.483% 4.003% 2.534% 5.681% 6.919% 2.710% 4.970% 5.312% 5.648% 4.152% 5.980% 4.984% 8.217% 2.055% 4.540% 8.076% 5.486% 11.981% 3.700% 7.093%
29095 26826 15203 13244 13552 57815 29397 11367 13929 16107 45704 23019 17270 14713 21180 47543 51711 26024 25197 17812 46400 60833 20491 32248 11659 122996 42178 27523 20538 29952 85576 40444
5.451% 1.760% 1.549% 3.899% 1.650% 4.098% 1.579% 4.132% 2.494% 2.592% 5.193% 8.478% 2.335% 4.067% 3.029% 4.245% 8.155% 1.780% 4.357% 4.900% 4.919% 3.850% 4.733% 5.193% 7.397% 4.593% 4.305% 6.919% 6.801% 10.762% 3.414% 8.504%
28840 26815 15343 13210 13771 57800 29399 11231 14004 16222 45226 22750 17205 14494 21330 47688 51381 25968 24974 17834 46124 60958 20342 31890 11487 120304 42200 27547 20118 29566 85775 40359
4.527% 1.718% 2.485% 3.632% 3.293% 4.071% 1.586% 2.886% 3.046% 3.325% 4.092% 7.210% 1.950% 2.518% 3.760% 4.563% 7.465% 1.560% 3.433% 5.029% 4.294% 4.063% 3.971% 4.025% 5.812% 2.304% 4.360% 7.012% 4.618% 9.334% 3.654% 8.277%
28848 26733 15197 13080 13628 57816 29261 11224 13856 16053 45824 23026 17193 14477 21249 47559 51024 25953 24868 17646 46316 60773 20356 31735 11469 119142 41926 27724 20307 29419 85636 39666
4.556% 1.407% 1.510% 2.612% 2.220% 4.100% 1.109% 2.822% 1.957% 2.248% 5.469% 8.511% 1.878% 2.398% 3.366% 4.280% 6.718% 1.502% 2.994% 3.922% 4.728% 3.747% 4.043% 3.520% 5.647% 1.316% 3.682% 7.699% 5.601% 8.790% 3.486% 6.417%
28965 26740 15083 13149 13575 57833 29460 11483 13944 16056 46689 22783 17120 14630 21209 48513 53069 26088 25218 17379 47612 60926 20671 31939 11411 119959 42075 28169 20424 28179 85665 41103
4.980% 1.434% 0.748% 3.154% 1.823% 4.130% 1.797% 5.194% 2.605% 2.268% 7.459% 7.366% 1.446% 3.480% 3.172% 6.372% 10.995% 2.030% 4.444% 2.350% 7.659% 4.008% 5.653% 4.185% 5.112% 2.010% 4.051% 9.428% 6.209% 4.205% 3.521% 10.273%
Avg. Gap (𝑛 < 251) X-n251-k28 X-n256-k16 X-n261-k13 X-n266-k58 X-n270-k35 X-n275-k28 X-n280-k17 X-n284-k15 X-n289-k60 X-n294-k50 X-n298-k31 X-n303-k21 X-n308-k13 X-n313-k71 X-n317-k53 X-n322-k28 X-n327-k20 X-n331-k15 X-n336-k84 X-n344-k43 X-n351-k40 X-n359-k29 X-n367-k17 X-n376-k94 X-n384-k52 X-n393-k38 X-n401-k29 X-n411-k19 X-n420-k130 X-n429-k61 X-n439-k37 X-n449-k29 X-n459-k26 X-n469-k138 X-n480-k70 X-n491-k59
38684 18839 26558 75478 35291 21245 33503 20226 95151 47161 34231 21736 25859 94043 78355 29834 27532 31102 139111 42050 25896 51505 22814 147713 65940 38260 66154 19712 107798 65449 36391 55233 24139 221824 89449 66483
Avg. Gap (251 ≤ 𝑛 < 501) X-n502-k39 X-n513-k21 X-n524-k153 X-n536-k96 X-n548-k50 X-n561-k42 X-n573-k30 X-n586-k159 X-n599-k92 X-n613-k62 X-n627-k43 X-n641-k35 X-n655-k131 X-n670-k130 X-n685-k75 X-n701-k44 X-n716-k35 X-n733-k159 X-n749-k98 X-n766-k71 X-n783-k48 X-n801-k40 X-n819-k171 X-n837-k142 X-n856-k95 X-n876-k59 X-n895-k37 X-n916-k207 X-n936-k151 X-n957-k87 X-n979-k58 X-n1001-k43
69226 24201 154593 94846 86700 42717 50673 190316 108451 59535 62164 63684 106780 146332 68205 81923 43373 136187 77269 114417 72386 73311 158121 193737 88965 99299 53860 329179 132715 85465 118976 72355
6.566% 41211 20400 28741 84617 38146 24688 36677 22474 104159 52769 37652 23556 28736 102253 82587 32593 30646 34734 152846 46619 29243 55778 26132 156857 73705 43533 71565 23869 122761 74261 41165 60162 29543 252031 101314 77536
6.532% 8.286% 8.220% 12.108% 8.090% 16.206% 9.474% 11.114% 9.467% 11.891% 9.994% 8.373% 11.126% 8.730% 5.401% 9.248% 11.310% 11.678% 9.873% 10.866% 12.925% 8.296% 14.544% 6.190% 11.776% 13.782% 8.179% 21.089% 13.881% 13.464% 13.119% 8.924% 22.387% 13.618% 13.265% 16.625%
11.529% 75711 34910 176491 109897 110984 55936 60884 226245 131035 77555 76776 83138 120771 183183 92701 92723 59383 175848 102208 132968 108577 92125 192102 231002 117243 114212 106062 387367 200816 126220 138987 132976
9.368% 44.250% 14.165% 15.869% 28.009% 30.946% 20.151% 18.879% 20.824% 30.268% 23.506% 30.548% 13.103% 25.183% 35.915% 13.183% 36.912% 29.122% 32.276% 16.213% 49.997% 25.663% 21.491% 19.235% 31.786% 15.018% 96.922% 17.677% 51.314% 47.686% 16.819% 83.783%
5.829% 41330 20559 28524 82048 38333 25021 36636 22583 102202 50886 37344 23263 28518 100620 83632 33497 30603 33636 149229 46947 28373 56165 25588 156546 73570 44638 71787 23139 116362 74158 42161 60015 29100 245581 100121 75226
6.840% 9.130% 7.403% 8.705% 8.620% 17.774% 9.351% 11.653% 7.410% 7.898% 9.094% 7.025% 10.283% 6.994% 6.735% 12.278% 11.154% 8.147% 7.273% 11.646% 9.565% 9.048% 12.159% 5.980% 11.571% 16.670% 8.515% 17.385% 7.944% 13.307% 15.856% 8.658% 20.552% 10.710% 11.931% 13.151%
10.616% 77033 32858 171734 106031 104240 53110 62033 212545 126654 73633 70744 71986 118758 168210 82607 89704 52170 156268 92403 130101 96432 87187 178856 230226 105763 114175 70363 374899 161700 124190 132651 89175
11.278% 35.771% 11.088% 11.793% 20.231% 24.330% 22.418% 11.680% 16.785% 23.680% 13.802% 13.036% 11.217% 14.951% 21.116% 9.498% 20.282% 14.745% 19.586% 13.708% 33.219% 18.928% 13.113% 18.834% 18.882% 14.981% 30.641% 13.889% 21.840% 45.311% 11.494% 23.246%
5.061% 40127 19994 28510 79832 37382 24187 36653 22154 100418 50637 37163 23442 28326 99564 80690 32658 29784 34048 146620 44914 28236 55122 25522 151975 70471 41552 69430 22849 117418 70164 39752 60634 27347 238904 95032 72618
3.730% 6.131% 7.350% 5.769% 5.925% 13.848% 9.402% 9.532% 5.535% 7.370% 8.565% 7.849% 9.540% 5.871% 2.980% 9.466% 8.180% 9.472% 5.398% 6.811% 9.036% 7.023% 11.870% 2.885% 6.871% 8.604% 4.952% 15.914% 8.924% 7.204% 9.236% 9.779% 13.290% 7.700% 6.242% 9.228%
4.772% 40359 20372 28833 80115 37674 24482 36081 22295 99739 49929 36993 23748 28913 98899 80542 33206 30953 34578 146707 45571 28059 55183 25534 151390 70611 42934 69875 23521 115012 70969 41149 61144 28267 237548 95466 71702
8.107% 71908 28542 174150 103242 100850 49133 56048 205654 116840 67545 67523 70631 112289 168829 77890 90580 49480 148581 85046 129866 82839 86121 174446 208669 98164 107477 64225 353039 162903 103089 129633 85852
3.874% 17.937% 12.651% 8.852% 16.321% 15.020% 10.607% 8.059% 7.735% 13.454% 8.621% 10.909% 5.159% 15.374% 14.200% 10.567% 14.080% 9.101% 10.065% 13.502% 14.441% 17.474% 10.324% 7.707% 10.340% 8.236% 19.244% 7.248% 22.746% 20.621% 8.957% 18.654%
4.330% 8.137% 8.566% 6.144% 6.752% 15.237% 7.695% 10.229% 4.822% 5.869% 8.069% 9.257% 11.810% 5.164% 2.791% 11.303% 12.426% 11.176% 5.460% 8.373% 8.353% 7.141% 11.923% 2.489% 7.084% 12.216% 5.625% 19.323% 6.692% 8.434% 13.075% 10.702% 17.101% 7.089% 6.727% 7.850%
8.889% 72655 29422 168181 102355 102318 50287 55353 204649 117784 69069 69361 73624 110657 161571 78473 92198 50605 146080 85325 127752 87562 94076 172387 209540 102312 109693 73280 351887 154847 108664 133201 92974
4.953% 21.573% 8.790% 7.917% 18.014% 17.721% 9.236% 7.531% 8.606% 16.014% 11.577% 15.608% 3.631% 10.414% 15.055% 12.542% 16.674% 7.264% 10.426% 11.655% 20.965% 28.325% 9.022% 8.157% 15.003% 10.467% 36.056% 6.898% 16.676% 27.144% 11.956% 28.497%
4.399% 40247 20086 28631 80247 37360 24347 35844 21916 100808 49980 36497 23295 28219 99511 81804 32959 30280 34200 149982 45044 28101 54884 25094 152539 70044 41929 69596 23198 115837 70546 39887 60812 27435 243469 95167 72031
4.040% 6.619% 7.805% 6.318% 5.862% 14.601% 6.987% 8.355% 5.945% 5.977% 6.620% 7.174% 9.125% 5.814% 4.402% 10.474% 9.980% 9.960% 7.815% 7.121% 8.515% 6.561% 9.994% 3.267% 6.223% 9.590% 5.203% 17.685% 7.457% 7.792% 9.606% 10.101% 13.654% 9.758% 6.392% 8.345%
8.237% 72108 28844 165183 102551 101100 48537 54571 208415 117440 67296 69915 72517 112598 160473 76817 91600 50073 146443 85320 126515 87027 90039 172578 208974 100818 108742 70673 358336 153747 105588 137728 90653
4.163% 19.185% 6.849% 8.123% 16.609% 13.624% 7.692% 9.509% 8.289% 13.035% 12.468% 13.870% 5.449% 9.664% 12.626% 11.812% 15.446% 7.531% 10.419% 10.573% 20.226% 22.818% 9.143% 7.865% 13.324% 9.509% 31.216% 8.857% 15.847% 23.522% 15.761% 25.287%
4.309% 40003 20061 28256 79480 36927 24127 35370 21619 100451 49318 36189 22937 27714 98667 80206 32047 29763 33203 146345 44563 27495 54125 24323 151129 68838 41394 68591 21621 114105 69282 39175 59136 26334 238868 94341 70501
3.410% 6.487% 6.394% 5.302% 4.636% 13.566% 5.573% 6.887% 5.570% 4.574% 5.720% 5.525% 7.174% 4.917% 2.362% 7.418% 8.103% 6.755% 5.200% 5.976% 6.175% 5.087% 6.614% 2.313% 4.395% 8.191% 3.684% 9.684% 5.851% 5.856% 7.650% 7.066% 9.093% 7.684% 5.469% 6.044%
6.295% 71726 26767 168238 100967 98583 47301 53624 204726 115132 65123 67301 69308 109726 161354 74689 88246 47683 144836 82650 124163 80255 84979 170347 205416 96653 105570 62387 350863 151238 100635 127697 82613
3.611% 10.603% 8.826% 6.454% 13.706% 10.731% 5.824% 7.572% 6.160% 9.386% 8.264% 8.831% 2.759% 10.266% 9.507% 7.718% 9.937% 6.351% 6.964% 8.518% 10.871% 15.916% 7.732% 6.028% 8.642% 6.315% 15.832% 6.587% 13.957% 17.750% 7.330% 14.177%
3.758% 39912 19838 27962 79578 36798 24185 35545 21697 98873 49614 36084 22879 27586 97780 80138 32089 29522 33068 144870 44438 27585 53857 24443 149853 69524 40981 68410 21851 113290 69491 38809 59033 26554 234044 93986 70411
3.174% 5.303% 5.287% 5.432% 4.270% 13.839% 6.095% 7.273% 3.912% 5.201% 5.413% 5.259% 6.679% 3.974% 2.276% 7.558% 7.228% 6.321% 4.140% 5.679% 6.522% 4.567% 7.140% 1.449% 5.435% 7.112% 3.410% 10.851% 5.095% 6.176% 6.645% 6.880% 10.005% 5.509% 5.072% 5.908%
5.913% 71271 27202 166429 100532 99247 47134 53974 201846 115224 65851 66603 69601 109505 158554 74523 87910 47292 142941 83014 123989 80501 84397 168486 205231 98082 105116 61973 347313 145801 101187 126903 82012
2.954% 12.400% 7.656% 5.995% 14.472% 10.340% 6.514% 6.058% 6.245% 10.609% 7.141% 9.291% 2.552% 8.352% 9.263% 7.308% 9.036% 4.959% 7.435% 8.366% 11.211% 15.122% 6.555% 5.933% 10.248% 5.858% 15.063% 5.509% 9.860% 18.396% 6.663% 13.347%
4.486% 40220 20087 28310 79363 37138 22405 35769 21605 99962 50028 36601 23312 28394 98535 79907 32746 30349 33794 147474 44850 27676 53808 24893 150506 70002 42105 69872 22508 116654 70129 40381 60116 26712 237471 94618 71793
3.971% 6.625% 6.597% 5.147% 5.234% 5.460% 6.764% 6.818% 5.056% 6.079% 6.924% 7.251% 9.803% 4.777% 1.981% 9.761% 10.232% 8.655% 6.012% 6.659% 6.874% 4.471% 9.113% 1.891% 6.160% 10.050% 5.620% 14.184% 8.215% 7.151% 10.964% 8.841% 10.659% 7.054% 5.779% 7.987%
7.079% 71599 28136 172677 102177 94022 49390 53807 204828 116627 67346 77841 70952 109546 164643 77686 91921 51944 145364 84975 130645 87629 80746 170573 206980 100106 113121 74729 352460 153139 110763 134360 95172
3.428% 16.260% 11.698% 7.729% 8.445% 15.621% 6.185% 7.625% 7.539% 13.120% 25.219% 11.413% 2.590% 12.513% 13.901% 12.204% 19.761% 6.739% 9.973% 14.183% 21.058% 10.142% 7.875% 6.836% 12.523% 13.920% 38.747% 7.072% 15.389% 29.600% 12.930% 31.535%
Avg. Gap (501 < 𝑛 ≤ 1001)
30.190%
18.918%
12.253%
14.199%
13.344%
9.191%
8.571%
13.555%
Avg. Gap
15.863%
11.693%
8.428%
9.230%
8.616%
6.586%
6.074%
8.322%
experiments, a greedy decoding was used with ×8 instance augmentation. The results, summarized in Table 11, show once again that FiLMMeD variations outperform their nonFiLMMeD counterparts across tha majority of instances. Furthermore, FiLMMeD-CaDA† displays the best average A. Corrêa et al.
performance, surpassing even the Dual-Branch DistanceAware (DBDA) method from Jiang et al. (2026), which was specifically designed for cross-distribution generalization.
Page 18 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP Table 12 Zero-shot generalization performance on unseen VRP variants. The best MTL results (among neural models) are shown in gray, and the models improved by our approach are underlined. HGS-PyVRP
OR-Tools
MTPOMO
MVMoE
RouteFinder
Variant
Obj.
Gap
Obj.
Gap
Obj.
Gap
Obj.
Gap
Obj.
Gap
VRPMB OVRPMB VRPMBL VRPMBTW OVRPMBL OVRPMBTW VRPMBLTW OVRPMBLTW MDVRP MDOVRP MDVRPB MDVRPL MDVRPTW MDOVRPTW MDOVRPB MDOVRPL MDVRPBL MDVRPBTW MDVRPLTW MDOVRPBL MDOVRPBTW MDOVRPLTW MDVRPBLTW MDOVRPBLTW MDVRPMB MDOVRPMB MDVRPMBL MDVRPMBTW MDOVRPMBL MDOVRPMBTW MDVRPMBLTW MDOVRPMBLTW
13.54 9.01 13.78 25.51 9.01 16.97 25.85 16.97 11.89 7.97 11.64 11.90 19.33 13.00 8.69 7.97 11.68 22.03 19.35 8.69 14.37 13.00 22.06 14.37 10.68 7.66 10.71 19.29 7.66 12.96 19.31 12.96
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
14.93 10.59 15.42 29.97 10.59 19.31 30.44 19.31 12.52 8.16 12.22 12.52 19.62 13.09 8.87 8.16 12.22 22.40 19.66 8.87 14.49 13.09 22.43 14.49 12.22 8.88 12.23 22.39 8.87 14.49 22.43 14.49
10.27% 17.54% 11.90% 17.48% 17.54% 13.78% 17.76% 13.78% 5.27% 2.33% 5.01% 5.24% 1.55% 0.74% 2.15% 2.33% 4.66% 1.69% 1.58% 2.13% 0.87% 0.70% 1.70% 0.86% 14.37% 15.83% 14.23% 16.12% 15.73% 11.79% 16.16% 11.79%
15.04 10.87 15.41 28.31 10.85 18.51 28.73 18.51 16.07 10.28 15.18 16.30 26.68 17.57 10.94 10.28 15.80 30.55 27.13 10.94 19.69 17.58 31.09 19.69 15.14 10.91 15.49 28.44 10.90 18.56 28.93 18.56
11.32% 20.65% 11.97% 11.06% 20.43% 9.08% 11.27% 9.12% 35.74% 29.06% 30.66% 37.58% 38.56% 35.67% 26.08% 29.07% 35.54% 39.23% 40.71% 26.11% 37.62% 35.70% 41.52% 37.64% 42.22% 42.57% 45.23% 48.01% 42.45% 43.63% 50.36% 43.65%
14.99 10.85 15.33 28.32 10.82 18.55 28.70 18.55 16.02 10.24 15.12 16.25 26.67 17.57 10.89 10.24 15.73 30.55 27.12 10.90 19.69 17.58 31.06 19.69 15.08 10.90 15.40 28.46 10.88 18.61 28.89 18.60
10.94% 20.42% 11.37% 11.10% 20.14% 9.33% 11.16% 9.30% 35.35% 28.59% 30.13% 37.17% 38.51% 35.68% 25.56% 28.60% 34.95% 39.22% 40.67% 25.66% 37.62% 35.74% 41.39% 37.61% 41.67% 42.41% 44.37% 48.12% 42.13% 44.04% 50.19% 43.95%
14.88 10.72 15.18 28.29 10.72 18.45 28.65 18.45 15.98 10.18 15.05 16.20 26.51 17.48 10.82 10.18 15.62 30.36 26.93 10.82 19.59 17.48 30.86 19.60 14.99 10.77 15.28 28.43 10.76 18.49 28.80 18.50
10.13% 19.02% 10.32% 10.87% 19.01% 8.68% 10.82% 8.69% 35.02% 27.82% 29.53% 36.76% 37.64% 34.96% 24.74% 27.84% 33.98% 38.36% 39.69% 24.78% 36.95% 34.95% 40.49% 36.96% 40.80% 40.67% 43.27% 47.93% 40.62% 43.14% 49.69% 43.17%
Avg. Gap
*
9.40%
32.18%
31.97%
These experiments indicate that our method has great crosssize and cross-distribution capabilities, despite being primarily designed for cross-problem generalization.
5.7. Results on unseen tasks To evaluate the zero-shot generalization capabilities of FiLMMeD-CaDA and FiLMMeD-CaDA†, we conducted further experiments across 32 unseen variants. Following the experimental settings of Berto et al. (2025) and Li et al. (2025), these evaluations were performed on 100-node instances. The test set includes mixed backhauls (MB) and multi-depots (MD), variants that appeared in the settings of Zhou et al. (2024a), but remain unseen to the training protocol established by Berto et al. (2025). The results are summarized in Table 12. Overall, FiLMMeD-CaDA and FiLMMeD-CaDA† consistently improved compared to their counterparts, obtaining a lower average gap. Furthermore, FiLMMeD-CaDA† displayed the best performance across all evaluated neural baselines, consolidating the zero-shot generalization of our method.
5.8. Performance of the PO algorithm on other models To further evaluate the effectiveness of the PO algorithm in the MTL domain, we retrained MTPOMO, MVMoE and RouteFinder (once again, the modern Transformer-based Encoder variant of RouteFinder) using PO and RL. The RLbased models were trained via the standard REINFORCE with shared baselines algorithm. Following the experimental A. Corrêa et al.
31.49%
CaDA Obj.
Gap
14.83 9.73% 10.58 17.38% 15.05 9.27% 28.40 11.42% 10.58 17.41% 18.57 9.44% 28.75 11.38% 18.57 9.43% 17.16 45.33% 9.74 22.31% 17.86 54.31% 17.14 45.04% 27.18 41.22% 17.23 33.02% 10.93 26.06% 9.74 22.26% 17.50 50.52% 30.88 40.84% 27.63 43.44% 10.93 25.99% 19.38 35.44% 17.22 32.99% 31.57 43.84% 19.38 35.44% 17.51 65.10% 10.78 40.93% 16.86 58.45% 29.38 52.98% 10.78 40.91% 18.70 44.76% 29.95 55.78% 18.70 44.78% 36.36%
FiLMMeD-CaDA
CaDA†
FiLMMeD-CaDA†
Obj.
Gap
Obj.
Gap
Obj.
Gap
14.65 10.51 14.97 28.28 10.51 18.56 28.62 18.55 14.30 9.84 14.44 14.37 24.12 15.93 10.53 9.93 14.47 27.54 24.24 10.65 17.75 15.93 27.62 17.78 14.24 10.42 14.17 26.24 10.52 17.19 26.26 17.20
8.37% 16.71% 8.71% 10.92% 16.72% 9.38% 10.86% 9.32% 20.69% 23.70% 24.38% 21.27% 25.25% 23.03% 21.43% 24.73% 24.15% 25.59% 25.73% 22.85% 24.09% 22.98% 25.79% 24.23% 33.86% 36.20% 32.82% 36.64% 37.47% 33.09% 36.57% 33.16%
14.62 10.47 14.93 28.20 10.47 18.40 28.51 18.40 13.99 10.52 13.99 14.07 23.28 15.96 11.21 10.51 14.21 26.60 23.32 11.20 17.77 15.97 26.62 17.77 13.78 10.97 13.85 24.69 10.97 16.66 24.64 16.66
8.15% 16.24% 8.40% 10.62% 16.24% 8.42% 10.43% 8.41% 18.06% 32.18% 20.48% 18.64% 20.75% 23.17% 29.30% 32.01% 21.86% 21.12% 20.82% 29.13% 24.10% 23.27% 21.04% 24.08% 29.41% 43.35% 29.77% 28.37% 43.42% 28.89% 27.92% 28.89%
14.60 10.39 14.95 28.16 10.39 18.48 28.48 18.48 14.03 10.25 14.09 14.08 22.83 15.14 11.01 10.24 14.22 25.68 22.86 11.01 16.59 15.14 25.75 16.59 13.89 10.78 13.96 24.08 10.78 15.86 24.02 15.86
7.99% 15.34% 8.53% 10.45% 15.37% 8.90% 10.32% 8.91% 18.43% 28.73% 21.34% 18.70% 18.38% 16.76% 26.93% 28.63% 21.96% 16.85% 18.38% 26.90% 15.71% 16.76% 17.04% 15.72% 30.52% 40.90% 30.83% 25.14% 40.83% 22.62% 24.68% 22.55%
23.46%
23.41%
21.01%
protocol of Berto et al. (2025), the average performance gaps are reported in Table 13. Across all three baselines, the PO algorithm exhibited superior generalization, underscoring even more its viability for future MTL VRP research. Moreover, its consistent performance improvements across different model architectures display great modularity, suggesting the application of PO for MTL is architecture-agnostic rather than tied to a specific model. Table 13 Average gaps across 16 VRP variants: RL vs. PO comparison. Model MTPOMO MVMoE RouteFinder
RL
PO
2.354% 2.169% 2.049%
2.141% 1.938% 1.807%
5.9. Complexity analysis In this section, we report the computational overhead of all FiLMMeD variations. The results, reported in Table 14, show that the FiLM mechanism adds fewer than 0.01M parameters, an increase of less than 1%. Similarly, the combined GPU memory overhead introduced by FiLM and PO remains negligible across all configurations. Additionally, the training times per epoch are nearly identical between FiLMMeD and its counterparts, indicating
Page 19 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP
that PO introduces minimal overhead compared to REINFORCE despite its 𝑂(𝑁 2 ) complexity in pairwise comparisons. This efficiency stems from the fact that the preference computation is lightweight relative to the autoregressive decoder’s forward pass. Table 14 Computational complexity analysis. Params (M)
Avg. GPU Memory (GB)
Avg. Training Epoch Time
MTPOMO FiLMMeD-MTPOMO
1.25 1.26
6.55 6.81
2m 5s 2m 9s
MVMoE FiLMMeD-MVMoE
3.68 3.68
9.00 9.49
3m 20s 3m 21s
CaDA FiLMMeD-CaDA CaDA† FiLMMeD-CaDA†
3.43 3.43 3.56 3.56
5.47 5.97 6.92 7.74
2m 50s 2m 49s 3m 34s 3m 38s
Model
6. Conclusion This paper introduces FiLMMeD, a novel MTL model focused on solving MDVRP variants. FiLMMeD augments the standard encoder architecture used in MTL works with a FiLM-based conditioning mechanism that adapts the model’s internal representations to the specific set of active constraints in each instance. We demonstrate that this mechanism effectively learns a more discriminative latent space of node embeddings, leading to a better generalization. We also propose a CL training strategy that progressively exposes the model to increasingly more complex MDVRP variants, based on the number of constraints. Compared to established training strategies and sampling regimens, our curriculum yielded substantially improved generalization across the 24 tested MDVRP variants. Finally, we explored the use of PO as an alternative to RL to train FiLMMeD on singledepot variants. Across different experimental settings, our results show that PO provides a faster convergence and better generalization, offering an early but compelling signal that PO may be the preferred training paradigm for future MTL routing works. We have confirmed the effectiveness of FiLMMeD through extensive experiments on both multiand single-depot problems. Additional ablation studies confirm the isolated impact of each individual contribution. For future work, we envision augmenting the PO algorithm to further improve the generalization of existing methods in the MTL setting.
CRediT authorship contribution statement Arthur Corrêa: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. Paulo Nascimento: Formal analysis, Project administration, Supervision, Writing - review & editing. Samuel Moniz: Funding acquisition, Project administration, Supervision, Writing - review & editing. A. Corrêa et al.
Acknowledgements This research is sponsored by national funds through FCT - Fundação para a Ciência e a Tecnologia (FCT), Portugal, through doctoral grant 2025.00622.BD and projects UID/00285/2025, and LA/P/0112/2020. This work was also supported by the European Union - NextGenerationEU, through the Portuguese Republic’s Recovery and Resilience Plan Partnership Agreement [project C64580887000000067], within the scope of the project PRODUTECH R3 – “Agenda Mobilizadora da Fileira das Tecnologias de Produção para a Reindustrialização”, Total project investment: 166.988.013,71 Euros; Total Grant : 97.111.730,27 Euros; and by the European Regional Development Fund (ERDF) through the Operational Program for Competitiveness and Internationalization (COMPETE 2020) under the project POCI-01-0247-FEDER-046102 (PRODUTECH4S&C).
References Bello, I., Pham, H., Le, Q.V., Norouzi, M., Bengio, S., 2017. Neural combinatorial optimization with reinforcement learning, in: 5th International Conference on Learning Representations, ICLR 2017 - Workshop Track Proceedings. Bengio, Y., Lodi, A., Prouvost, A., 2021. Machine learning for combinatorial optimization: A methodological tour d’horizon. European Journal of Operational Research 290, 405–421. URL: https:// www.sciencedirect.com/science/article/pii/S0377221720306895, doi:10. 1016/j.ejor.2020.07.063. Berto, F., Hua, C., Zepeda, N.G., Hottung, A., Wouda, N., Lan, L., Park, J., Tierney, K., Park, J., 2025. RouteFinder: Towards Foundation Models for Vehicle Routing Problems. Transactions on Machine Learning Research URL: https://openreview.net/forum?id=QzGLoaOPiY. Bi, J., Ma, Y., Zhou, J., Song, W., Cao, Z., Wu, Y., Zhang, J., 2025. Learning to handle complex constraints for vehicle routing problems, in: Advances in Neural Information Processing Systems. Bogyrbayeva, A., Meraliyev, M., Mustakhov, T., Dauletbayev, B., 2024. Machine learning to solve vehicle routing problems: A survey. Trans. Intell. Transport. Sys. 25, 4754–4772. URL: https://doi.org/10.1109/ TITS.2023.3334976, doi:10.1109/TITS.2023.3334976. Braekers, K., Ramaekers, K., Van Nieuwenhuyse, I., 2016. The vehicle routing problem: State of the art classification and review. Computers & Industrial Engineering 99, 300–313. URL: https: //www.sciencedirect.com/science/article/pii/S0360835215004775, doi:https://doi.org/10.1016/j.cie.2015.12.007. Cattaruzza, D., Absi, N., Feillet, D., González-Feliu, J., 2017. Vehicle routing problems for city logistics. EURO Journal on Transportation and Logistics 6, 51–79. doi:10.1007/s13676-014-0074-0. Chalumeau, F., Surana, S., Bonnet, C., Grinsztajn, N., Pretorius, A., Laterre, A., Barrett, T.D., 2023. Combinatorial optimization with policy adaptation using latent space search, in: Advances in Neural Information Processing Systems. Chen, X., Tian, Y., 2019. Learning to perform local rewriting for combinatorial optimization, in: Advances in Neural Information Processing Systems. Cheng, H., Zheng, H., Cong, Y., Jiang, W., Pu, S., 2023. Select and optimize: Learning to solve large-scale tsp instances, in: Ruiz, F., Dy, J., van de Meent, J.W. (Eds.), Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR. pp. 1219– 1231. URL: https://proceedings.mlr.press/v206/cheng23a.html. Cordeau, J.F., Gendreau, M., Laporte, G., 1997. A tabu search heuristic for periodic and multi-depot vehicle routing problems. Networks 30, 105–119.
Page 20 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP Corrêa, A., Silva, C., Xu, L., Brintrup, A., Moniz, S., 2026. Tunensearch: A hybrid transfer learning and local search approach for solving vehicle routing problems. Computers & Operations Research 190, 107433. URL: https://www.sciencedirect.com/science/article/pii/ S0305054826000511, doi:https://doi.org/10.1016/j.cor.2026.107433. Crevier, B., Cordeau, J.F., Laporte, G., 2007. The multi-depot vehicle routing problem with inter-depot routes. European Journal of Operational Research 176, 756–773. URL: https: //www.sciencedirect.com/science/article/pii/S0377221705006983, doi:https://doi.org/10.1016/j.ejor.2005.08.015. Drakulic, D., Michel, S., Mai, F., Sors, A., Andreoli, J.M., 2023. BQ-NCO: Bisimulation quotienting for efficient neural combinatorial optimization, in: Thirty-seventh Conference on Neural Information Processing Systems. URL: https://openreview.net/forum?id=BRqlkTDvvm. Elatar, S., Abouelmehdi, K., Riffi, M.E., 2023. The vehicle routing problem in the last decade: variants, taxonomy and metaheuristics. Procedia Computer Science 220, 398–404. URL: https://www.sciencedirect. com/science/article/pii/S1877050923005860, doi:10.1016/j.procs.2023. 03.051. the 14th International Conference on Ambient Systems, Networks and Technologies Networks (ANT) and The 6th International Conference on Emerging Data and Industry 4.0 (EDI40). Fu, Z.H., Qiu, K.B., Zha, H., 2021. Generalize a small pre-trained model to arbitrarily large tsp instances, in: Proceedings of the AAAI conference on artificial intelligence, pp. 7474–7482. Goh, Y.L., Cao, Z., Ma, Y., Zhou, J., Dupty, M.H., Lee, W.S., 2025. SHIELD: Multi-task multi-distribution vehicle routing solver with sparsity and hierarchy, in: Forty-second International Conference on Machine Learning. URL: https://openreview.net/forum?id=6DJEaz1cCj. Grinsztajn, N., Furelos-Blanco, D., Surana, S., Bonnet, C., Barrett, T.D., 2023. Winner takes it all: Training performant rl populations for combinatorial optimization, in: Advances in Neural Information Processing Systems. Ha, D., Dai, A.M., Le, Q.V., 2017. Hypernetworks, in: International Conference on Learning Representations. URL: https://openreview. net/forum?id=rkpACe1lx. Hottung, A., Tierney, K., 2020. Neural large neighborhood search for the capacitated vehicle routing problem, in: European Conference on Artificial Intelligence, pp. 443–450. doi:10.3233/FAIA200124. Hou, Q., Yang, J., Su, Y., Wang, X., Deng, Y., 2023. Generalize learned heuristics to solve large-scale vehicle routing problems in real-time, in: The Eleventh International Conference on Learning Representations. Huang, Z., Zhou, J., Cao, Z., XU, Y., 2025. Rethinking light decoder-based solvers for vehicle routing problems, in: The Thirteenth International Conference on Learning Representations. URL: https://openreview. net/forum?id=4pRwkYpa2u. Hudson, B., Li, Q., Malencia, M., Prorok, A., 2022. Graph neural network guided local search for the traveling salesperson problem, in: ICLR 2022 - 10th International Conference on Learning Representations. Jiang, Y.B., Tao, D.Y., Chen, W.J., Xu, F.R., 2026. Dual-branch distance-aware neural solver for cross-distribution vehicle routing problems. Neurocomputing 679, 133239. URL: https: //www.sciencedirect.com/science/article/pii/S0925231226006363, doi:https://doi.org/10.1016/j.neucom.2026.133239. Kim, M., Park, J., Kim, J., 2021. Learning collaborative policies to solve nphard routing problems, in: Advances in Neural Information Processing Systems, pp. 10418–10430. Kim, M., Park, J., Park, J., 2022. Sym-nco: Leveraging symmetricity for neural combinatorial optimization, in: Advances in Neural Information Processing Systems. Kool, W., van Hoof, H., Welling, M., 2019. Attention, learn to solve routing problems!, in: 7th International Conference on Learning Representations, ICLR 2019. Kwon, Y.D., Choo, J., Kim, B., Yoon, I., Gwon, Y., Min, S., 2020. Pomo: Policy optimization with multiple optima for reinforcement learning, in: Advances in Neural Information Processing Systems. Lei, K., Guo, P., Wang, Y., Wu, X., Zhao, W., 2022. Solve routing problems with a residual edge-graph attention neural network. Neurocomputing 508, 79–98. URL: https://www.sciencedirect.com/science/article/
A. Corrêa et al.
pii/S092523122200978X, doi:https://doi.org/10.1016/j.neucom.2022.08. 005.
Li, H., Liu, F., Zheng, Z., Zhang, Y., Wang, Z., 2025. CaDA: Crossproblem routing solver with constraint-aware dual-attention, in: Fortysecond International Conference on Machine Learning. URL: https: //openreview.net/forum?id=CS4RyQuTig. Li, J., Tian Dai, B., Niu, Y., Xiao, J., Wu, Y., 2024. Multi-type attention for solving multi-depot vehicle routing problems. IEEE Transactions on Intelligent Transportation Systems 25, 17831–17840. doi:10.1109/TITS. 2024.3413077. Li, S., Yan, Z., Wu, C., 2021. Learning to delegate for large-scale vehicle routing. Advances in Neural Information Processing Systems 34, 26198–26211. Li, Y., Chu, F., Feng, C., Chu, C., Zhou, M.C., 2019. Integrated production inventory routing planning for intelligent food logistics systems. IEEE Transactions on Intelligent Transportation Systems 20, 867–878. doi:10. 1109/TITS.2018.2835145. Liao, Z., Chen, J., Wang, D., Zhang, Z., Wang, J., 2025. BOPO: Neural combinatorial optimization via best-anchored and objective-guided preference optimization, in: Forty-second International Conference on Machine Learning. URL: https://openreview.net/forum?id=FLy6yXdrlW. Liu, F., Lin, X., Wang, Z., Zhang, Q., Xialiang, T., Yuan, M., 2024. Multi-task learning for routing problem with cross-problem zero-shot generalization, in: Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1898–1908. doi:10.1145/3637528.3672040. Liu, S., Cao, Z., Feng, S., Ong, Y.S., 2025. A mixed-curvature based pre-training paradigm for multi-task vehicle routing solver, in: Fortysecond International Conference on Machine Learning. URL: https: //openreview.net/forum?id=JsPyLqCgks. Lyu, W., Wang, H., Hong, Z., Wang, G., Yang, Y., Liu, Y., Zhang, D., 2023. Rede: Exploring relay transportation for efficient last-mile delivery, in: 2023 IEEE 39th International Conference on Data Engineering (ICDE), pp. 3003–3016. doi:10.1109/ICDE55515.2023.00230. Ma, Y., Cao, Z., Chee, Y.M., 2023. Learning to search feasible and infeasible regions of routing problems with flexible neural k-opt, in: Advances in Neural Information Processing Systems. Ma, Y., Li, J., Cao, Z., Song, W., Zhang, L., Chen, Z., Tang, J., 2021. Learning to iteratively solve routing problems with dual-aspect collaborative transformer, in: Advances in Neural Information Processing Systems, pp. 11096–11107. van der Maaten, L., Hinton, G., 2008. Visualizing data using t-SNE. Journal of Machine Learning Research 9, 2579–2605. URL: http://www.jmlr. org/papers/v9/vandermaaten08a.html. Mazyavkina, N., Sviridov, S., Ivanov, S., Burnaev, E., 2021. Reinforcement learning for combinatorial optimization: A survey. doi:10.1016/j.cor. 2021.105400. Nazari, M., Oroojlooy, A., Takáč, M., Snyder, L.V., 2018. Reinforcement learning for solving the vehicle routing problem, in: Advances in Neural Information Processing Systems, pp. 9839–9849. Pan, M., Lin, G., Luo, Y.W., Zhu, B., Dai, Z., Sun, L., Yuan, C., 2025a. Preference optimization for combinatorial optimization problems, in: Forty-second International Conference on Machine Learning. URL: https://openreview.net/forum?id=Jwe5FJ8QGx. Pan, Y., Cao, Z., GU, C., Liu, L., Zhao, P., Chen, Y., Lin, F., 2025b. Multitask vehicle routing solver via mixture of specialized experts under statedecomposable MDP, in: The Thirty-ninth Annual Conference on Neural Information Processing Systems. URL: https://openreview.net/forum? id=ezSyZM6Lp7. Perez, E., Strub, F., de Vries, H., Dumoulin, V., Courville, A., 2018. Film: visual reasoning with a general conditioning layer, in: Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence and Thirtieth Innovative Applications of Artificial Intelligence Conference and Eighth AAAI Symposium on Educational Advances in Artificial Intelligence, AAAI Press. Pessoa, A., Sadykov, R., Uchoa, E., Vanderbeck, F., 2020. A generic exact solver for vehicle routing and related problems. Mathematical Programming 183, 483–523. doi:10.1007/s10107-020-01523-z.
Page 21 of 22
FiLMMeD: Feature-wise Linear Modulation for Cross-Problem MDVRP Pillac, V., Gendreau, M., Guéret, C., Medaglia, A.L., 2013. A review of dynamic vehicle routing problems. European Journal of Operational Research 225, 1–11. URL: https: //www.sciencedirect.com/science/article/pii/S0377221712006388, doi:https://doi.org/10.1016/j.ejor.2012.08.015. Ramos, T.R.P., Gomes, M.I., Barbosa-Póvoa, A.P., 2020. A new matheuristic approach for the multi-depot vehicle routing problem with inter-depot routes. OR Spectrum 42, 75–110. URL: https://doi.org/10.1007/ s00291-019-00568-7, doi:10.1007/s00291-019-00568-7. Roberto, P., Costa, O.D., Rhuggenaath, J., Zhang, Y., Akcay, A., 2020. Learning 2-opt heuristics for the traveling salesman problem via deep reinforcement learning, in: Proceedings of Machine Learning Research, pp. 465–480. Shazeer, N., 2020. Glu variants improve transformer. URL: https://arxiv. org/abs/2002.05202, arXiv:2002.05202. Uchoa, E., Pecin, D., Pessoa, A., Poggi, M., Vidal, T., Subramanian, A., 2017. New benchmark instances for the capacitated vehicle routing problem. European Journal of Operational Research 257, 845–858. doi:10.1016/j.ejor.2016.08.012. Vaswani, A., Shazeer, N.M., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, L., Polosukhin, I., 2017. Attention is all you need, in: Advances in Neural Information Processing Systems, pp. 5999–6009. Vinyals, O., Fortunato, M., Jaitly, N., 2015. Pointer networks, in: Advances in Neural Information Processing Systems, pp. 2692–2700. Wang, Y., Jia, Y.H., Chen, W.N., Mei, Y., 2025. Soft parameter sharing model for cross-problem generalization in vehicle routing problems. IEEE Transactions on Artificial Intelligence , 1–15doi:10.1109/TAI. 2025.3576336. Wang, Z., Sheu, J.B., 2019. Vehicle routing problem with drones. Transportation Research Part B: Methodological 122, 350– 364. URL: https://www.sciencedirect.com/science/article/pii/ S0191261518307884, doi:10.1016/j.trb.2019.03.005. Wouda, N.A., Lan, L., Kool, W., 2024. PyVRP: a high-performance VRP solver package. INFORMS Journal on Computing 36, 943–955. URL: https://doi.org/10.1287/ijoc.2023.0055, doi:10.1287/ijoc.2023.0055. Wu, Y., Song, W., Cao, Z., Zhang, J., Lim, A., 2022. Learning improvement heuristics for solving routing problems. IEEE Transactions on Neural Networks and Learning Systems 33, 5057–5069. doi:10.1109/TNNLS. 2021.3068828. Ye, H., Wang, J., Liang, H., Cao, Z., Li, Y., Li, F., 2024. Glop: Learning global partition and local construction for solving large-scale routing problems in real-time, in: Proceedings of the AAAI Conference on Artificial Intelligence. Yu, T., Kumar, S., Gupta, A., Levine, S., Hausman, K., Finn, C., 2020. Gradient surgery for multi-task learning, in: Proceedings of the 34th International Conference on Neural Information Processing Systems, Curran Associates Inc., Red Hook, NY, USA. Zhang, B., Sennrich, R., 2019. Root mean square layer normalization. Curran Associates Inc., Red Hook, NY, USA. Zhang, Y., Yang, Q., 2022. A survey on multi-task learning. IEEE Transactions on Knowledge and Data Engineering 34, 5586–5609. doi:10.1109/TKDE.2021.3070203. Zheng, Y., Luo, F., Wang, Z., Wu, Y., Zhou, Y., 2025. MTL-KD: Multitask learning via knowledge distillation for generalizable neural vehicle routing solver, in: The Thirty-ninth Annual Conference on Neural Information Processing Systems. URL: https://openreview.net/forum? id=rlH3e7VlY8. Zheng, Z., Zhou, C., Xialiang, T., Yuan, M., Wang, Z., 2024. Udc: A unified neural divide-and-conquer framework for large-scale combinatorial optimization problems. Advances in Neural Information Processing Systems 37, 6081–6125. Zhou, J., Cao, Z., Wu, Y., Song, W., Ma, Y., Zhang, J., Xu, C., 2024a. Mvmoe: Multi-task vehicle routing solver with mixture-of-experts. Proceedings of Machine Learning Research , 61804–61824. Zhou, J., Wu, Y., Cao, Z., Song, W., Zhang, J., Shen, Z., 2024b. Collaboration! towards robust neural methods for routing problems, in: Advances in Neural Information Processing Systems, pp. 121731–121764.
A. Corrêa et al.
Zhou, J., Wu, Y., Song, W., Cao, Z., Zhang, J., 2023. Towards omnigeneralizable neural methods for vehicle routing problems, in: Proceedings of Machine Learning Research, pp. 42769–42789. Zong, Z., Wang, H., Wang, J., Zheng, M., Li, Y., 2022. Rbg: Hierarchically solving large-scale routing problems in logistic systems via reinforcement learning, in: Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 4648–4658.
Page 22 of 22