Title: Neural Combinatorial Optimization for Time-Dependent Traveling Salesman Problem
Venue: NeurIPS 2025
Accepted: yes
Keywords: Time-Dependent Routing

Neural Combinatorial Optimization

Traveling Salesman Problem

Deep Reinforcement Learning
Abstract: The Time-Dependent Traveling Salesman Problem (TDTSP) extends the classical TSP by allowing dynamic edge weights that vary with departure time, reflecting real-world scenarios such as transportation networks, where travel times fluctuate due to congestion patterns. TDTSP violates symmetry, triangle inequality, and cyclic invariance properties of classical TSP, creating unique computational challenges.
In this paper, we propose a neural model that extends MatNet from static asymmetric TSP to time-dependent settings by using an adjacency tensor to capture temporal variations, followed by a time-aware decoder. Our architecture addresses the unique challenge of asymmetry and triangle inequality violations that change dynamically over time.
Beyond architectural innovations, our research reveals a critical evaluation insight: many practical TDTSP instances maintain the same optimal solution regardless of time-dependent edge weights. This exposes a fundamental limitation in current evaluation practices for TDTSP that rely solely on average travel time metrics across all instances. Such metrics fail to effectively distinguish between methods that genuinely capture temporal dynamics and those that merely perform well on static routing problems. 
Instead, we present extensive experiments on real-world datasets, evaluating our approach on both entire datasets and specifically filtered instances where temporal dependencies alter the optimal solution. Results show that our method achieves state-of-the-art average optimality gap on full instances and significant travel-time reduction on instances where time-aware routing saves time. These results demonstrate state-of-the-art ability to identify and exploit temporal dependencies, setting new standards for evaluating time-dependent routing problems.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper develops a neural combinatorial optimization method for time dependent travelling salesman problem. This extends previous works on neural combinatorial optimization methods for TSP, adding a time dependent component. The architecture extends MatNet to incorporate temporal information. The paper also shows that only a small fraction of instances from real-world distribution show substantial improvement when taking temporal information into account and propose selecting between the best solution found using only static information with that found using temporal information and performs further local search.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The work is competently done.\n- The paper is clearly written.\n- The analysis showing where the improvements are coming from is useful. In cases where temporal information gives more than 3% gain, the method does show useful improvements.\n\nWeaknesses:\n- Novelty is not that high. The method appears to be a straightforward adaptation of MatNet to incorporate temporal information.\n- On average, results are similar to MatNet. Results are only better by selecting the better of the two methods and doing additional local search.\n\nQUESTIONS:\n1. For figure 2, what are the two lines shown and the region between the lines? Is that the confidence interval and if it is, is it one or two standard deviation?\n2. For comparison with simulated annealing, is the SA implementation also parallelized with GPU for fair comparison?\n3. For question 7 of the checklist, \"Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?\", you answered \"Answer: [No]. Justification: Error bars do not support our conclusion.\" Is it not possible to increase the test sample size so that we can trust your experimental conclusions?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the Time-Dependent Traveling Salesman Problem (TDTSP), a more realistic version of the Traveling Salesman Problem where travel times on edges are not static but change based on the departure time. The authors highlight that TDTSP violates fundamental properties of classical TSP like symmetry and triangle inequality, making it computationally challenging. Existing neural combinatorial optimization methods often fail to capture both spatial and temporal dynamics simultaneously, or their evaluation methodologies are insufficient for TDTSP.\n\nThe key insight of the paper is an empirical analysis revealing that in many practical TDTSP instances, the optimal solution (the sequence of visited nodes) remains the same regardless of time-dependent edge weights. This exposes a limitation in prior evaluation methods that rely solely on average tour duration, as such metrics cannot effectively differentiate between methods truly exploiting temporal dynamics and those merely performing well on static routing.\n\nTo overcome these challenges, the paper proposes:\n\nA novel end-to-end neural network model designed to directly encode the time-dependent adjacency tensor, thereby capturing complex spatio-temporal dynamics simultaneously. The model uses an encoder-decoder architecture with dual graph attention layers in the encoder and a time-aware decoder that incorporates sinusoidal embeddings for temporal information.\n\nAn effective inference process that enhances solution quality based on the data distribution, specifically employing a Mixture of Experts (MoE) approach (combining their neural model with a state-of-the-art Asymmetric TSP (ATSP) solver) and a local search refinement.\n\nA new evaluation methodology that accounts for instances where temporal dependencies truly alter the optimal solution, rather than just relying on average performance across all instances.\n\nThe paper presents extensive experiments on real-world datasets from cities like Beijing, Lyon, Nairobi, and London, demonstrating state-of-the-art performance in terms of average optimality gap and significant travel time reduction on instances where time-aware routing provides actual benefits. This establishes new standards for evaluating routing problems with temporal dependencies.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nQuality: The paper demonstrates a high level of technical quality. The proposed neural architecture for encoding time-dependent adjacency tensors is well-designed to capture spatio-temporal dynamics. The integration of sinusoidal embeddings and the inference process with Mixture of Experts (MoE) and local search show a sophisticated understanding of both neural network design and combinatorial optimization heuristics. The experimental setup is robust, comparing against various baselines (exact methods, ATSP solvers, heuristics) on real-world datasets from multiple cities, which strengthens the validity of their claims. The results indicate state-of-the-art performance, especially in instances where time-awareness is critical.\n\nClarity: The paper is generally well-structured and easy to follow. The problem statement for TDTSP is clearly defined. The proposed model's architecture (encoder-decoder, dual graph attention, time-aware decoder) is explained with good accompanying diagrams (Figure 3). The rationale behind each component, such as the three-stage process for the encoder's update function, is clearly articulated. The empirical analysis of TDTSP data distribution (Pareto distribution) and its implications for evaluation are presented concisely and are supported by Figure 2.\n\nSignificance: The paper makes several significant contributions.\nNovel Evaluation Methodology: The identification of the \"Pareto distribution\" in real-world TDTSP instances and the proposal of a new evaluation method that focuses on instances where temporal dependencies truly matter is a critical and highly significant contribution. This insight can re-shape how TDTSP algorithms are assessed in future research.\n\nEffective Model for TDTSP: The proposed neural model effectively tackles the complex spatio-temporal nature of TDTSP, which has been a challenge for existing neural methods. Its state-of-the-art performance on real-world datasets demonstrates its practical value.\n\nPractical Relevance: TDTSP has immense practical applications in logistics and transportation, where even small improvements can lead to substantial cost savings. By providing a more effective and accurately evaluated solution, the paper has a high potential for real-world impact.\n\nOriginality: The originality of the paper lies primarily in two areas:\nEmpirical Data Analysis and New Evaluation: The discovery of the Pareto distribution of savings in real-world TDTSP instances and the subsequent proposal for a refined evaluation methodology is a truly original insight. This challenges existing benchmarks and provides a more meaningful way to assess performance.\n\nDirect Encoding of Time-Dependent Tensor: While the base encoder-decoder framework and dual graph attention layers build upon prior work (e.g., MatNet, Attention Model) , the specific approach to directly encode the higher-dimensional time-dependent adjacency tensor and integrate temporal information throughout the attention mechanism is novel. The use of MoE in inference to leverage the ATSP solution for \"static\" instances is also a smart and original engineering contribution.\n\n\nWeaknesses:\n\nClarity on \"Online Refinement\": While the paper mentions \"online refinement methods\", the term \"online\" might be misleading if it refers strictly to inference-time enhancements. It should be clarified if the MoE and local search components involve any real-time adaptation or merely post-processing of the neural model's output. The current description implies they are sequential post-processing steps.\n\nLimited Deep Dive into Model Sensitivity: The paper does not thoroughly explore the sensitivity of the model to different hyperparameters (e.g., number of dual graph attention layers (N), learning rate) or variations in the sinusoidal embedding. Such analysis would provide more insights into the model's robustness.\n\nScalability for Larger Instances (my major concern): While the paper shows good scalability up to 50 nodes for Beijing, and 20 nodes for other cities, TDTSP in real-world logistics can involve hundreds or thousands of nodes. The computational time for \"Ours (Imp)\" on Beijing-50 (1549.88s) is quite high, suggesting that further work on scalability for very large instances (at least 100) is needed, or a discussion of trade-offs.\n\nTraining Complexity/Algorithm: The paper states that it cannot use the POMO training framework and instead relies on the REINFORCE algorithm. While REINFORCE is a classic choice, it is known for high variance. A brief discussion on how training stability or convergence was managed, or whether alternative lower-variance policy gradient methods were considered, would be beneficial.\n\nQUESTIONS:\nQuestion: Could the authors provide more insights into the robustness of the proposed neural model? For example, how sensitive are the results to variations in the number of dual graph attention layers (N) or different initialization strategies?\nSuggestion: Conduct ablation studies or sensitivity analyses on key hyperparameters (e.g., varying the number of attention layers, exploring different learning rates for REINFORCE, or alternative temporal encoding methods beyond sinusoidal embeddings).\n\nQuestion: For real-world logistics, TDTSP instances can involve hundreds or thousands of nodes. The current experiments primarily focus on N=10, 20, and 50 nodes. How do the authors envision scaling this approach to much larger instances, given the reported inference times for N=50 (e.g., ~1550s for Beijing-50)?\n\nQuestion: The paper mentions using the REINFORCE algorithm for training due to the asymmetric traveling times. REINFORCE is known for high variance. Can the authors elaborate on how they managed training stability (e.g., specific baselines, large batch sizes)? Were other policy gradient methods (e.g., Actor-Critic, PPO) considered, and if so, what were the reasons for choosing REINFORCE?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a neural model that encodes the time-dependent adjacency tensor to address asymmetry and triangle inequality violations. Beyond architectural innovations, the research highlights a critical evaluation insight: many practical TDTSP instances retain the same optimal solution despite time-dependent edge weights, exposing a fundamental flaw in current evaluation practices that rely solely on average travel time metrics.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n\nThe paper identifies critical limitations in the evaluation of existing algorithms and extends prior methodologies, with experimental results demonstrating notable advantages. Additionally, the logical structure of the paper is well-organized, facilitating clear comprehension of the research framework and conclusions.\n\nWeaknesses:\n\n1.\tThe paper proposes a time-dependent adjacency tensor encoding and time-aware decoder for TDTSP, but the abstract’s claim to address asymmetry and triangle inequality requires clarification. While the authors frame this as a novel contribution, Kwon et al. [20] already resolved these issues for static ATSP using matrix encoding. The extension to TDTSP is valuable, but the abstract must explicitly acknowledge the foundational work in [20] and highlight how the temporal dimension introduces unique innovations (e.g., tensor-based spatiotemporal integration). Without this distinction, the novelty of addressing asymmetry is diminished.\n\n2.\tThe model’s ability to \"simultaneously capture spatiotemporal structures\" needs explicit comparison with prior works (e.g., [8,16,38]). Current neural approaches for TDTSP either separate temporal and spatial processing (e.g., [16] by node, [8] by time) or focus on static ATSP ([20]). To validate the claimed advantage, the authors should: 1) Provide a comparative diagram in the Introduction/Methodology showing how their tensor encoding differs from matrix-based methods in [20] and temporal segmentation in [8,16]. 2) Quantify the mechanism through which the tensor structure models time-space interactions (e.g., via attention layers that process temporal and spatial features jointly). \n\n3.\tThis paper describes the three-stage process in line 181, but Fig. 3 is not clearly labeled. It is recommended to modify Fig. 3.\n\n4.\tThe experimental comparison is limited to classical algorithms (DP, GR, SA, ACO) and static ATSP methods (MatNet), omitting state-of-the-art TDTSP-specific neural solvers. It is recommended to add benchmarks against SOTA TDTSP methods.\n\n5.\tThe experimental validation is currently limited to test sets of up to 50 nodes, which raises concerns about the algorithm's scalability to problem sizes exceeding 50. It is recommended to include additional experimental evaluations on larger-scale instances to verify the method's performance and efficiency in real-world applications with higher node counts.\n\nQUESTIONS:\nSee strengths and weaknesses.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the TDTSP, a generalization of the classic TSP where edge costs vary with departure time. The authors propose a novel neural model that directly encodes a time-dependent adjacency tensor using a dual-graph attention mechanism, and introduce a new evaluation protocol to enhance the solution quality based on the data distribution.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n1. This work addresses a practically significant and societally relevant problem, time-dependent route optimization in urban logistics, by advancing neural methods to better capture real-world spatiotemporal dynamics.\n2. This paper proposes to capture spatio-temporal dependencies more effectively than prior approaches that separate node and time representations, and integrates MoE with classical ATSP solvers and local search, leading to better empirical performance.\n3. The paper is written with a well-structured motivation and methodology, making it easy to understand.\nWeaknesses:\n1.\tThe setting and implementation of the experiment in this paper are not clear enough. For example, the setting and comparison details of the baseline should be given in the supplementary materials instead of the pseudo code of the algorithm. The reader concern about how to make a fair comparison with different types of baselines, rather than the principles of different algorithms.\n2.\tReinforcement learning design is under-specified. While REINFORCE is employed, the paper omits details on the reward signal, whether techniques like baselines or advantages are used to reduce variance, and how sparsity or instability is addressed.\n3.\tLack of robustness analysis. The paper does not report error bars or statistical significance testing, which weakens the reliability of the reported improvements, especially when differences are small.\n4.\tMissing ablation study. The individual contributions of key components (such as temporal embeddings, dual-graph attention, MoE, and local search) are not given, making it hard to assess their necessity or impact.\n5.\tMissing generalization analysis. The paper lacks analysis on the model’s generalization ability. It is unclear whether cross-city generalization or evaluation on perturbed/synthetic data was performed to test robustness.\n6.\tMinor issues:\nThe paper should clarify how keys and values for attention are computed (line 181); at minimum, this should be included in the supplementary material.\nIn Equation (5), the adjacency matrix should be indexed by time, i.e., A[t].\nSeveral notations are insufficiently explained, which affects readability and clarity.\n\nQUESTIONS:\n1. Clarification on Local Search Design: Could the authors specify which local search operators (e.g., 2-opt, relocation) were applied across different datasets or instance sizes? Additionally, how was the iteration limit k=2 determined? Providing this information would enhance the reproducibility and clarity of the proposed post-processing step.\n2. Definition and Justification of “Sensitive” Instances: Table 3 introduces a 3% optimality gap threshold to define \"sensitive\" instances. Is this threshold empirically derived, or dataset-specific? An ablation study or justification for this choice would help clarify its robustness and generalizability.\n3. Scalability of the Proposed Encoder: Can the proposed encoder architecture scale to larger graphs? If so, is any subgraph sampling or other approximation strategy employed? It would be helpful to report GPU memory usage and runtime as a function of graph size n to assess scalability.\n4. Consideration of Alternative Policy Gradient Methods: Given the known limitations of REINFORCE in terms of spatial learning and sample efficiency, have the authors considered more stable or efficient policy gradient variants?\n\nLIMITATIONS:\nYES", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}