Title: Solving the Asymmetric Traveling Salesman Problem via Trace-Guided Cost Augmentation
Venue: NeurIPS 2025
Accepted: yes
Keywords: TSP

Combinatorial Optimization
Abstract: The Asymmetric Traveling Salesman Problem (ATSP) ranks among the most fundamental and notoriously difficult problems in combinatorial optimization. We propose a novel continuous relaxation framework for the Asymmetric Traveling Salesman Problem (ATSP) by leveraging differentiable constraints that encourage acyclic structures and valid permutations. Our approach integrates a differentiable trace-based Directed Acyclic Graph (DAG) constraint with a doubly stochastic matrix relaxation of the assignment problem, enabling gradient-based optimization over soft permutations. We develop a projected exponentiated gradient method with adaptive step size to minimize tour cost while satisfying the relaxed constraints. To recover high-quality discrete tours, we introduce a greedy post-processing procedure that iteratively corrects subtours using cost-aware cycle merging. Our method achieves state-of-the-art performance on standard asymmetric TSP benchmarks and demonstrates competitive scalability and accuracy, particularly on large or asymmetric instances where heuristic solvers such as LKH-3 struggle.

Solving Asymmetric Traveling Salesman Problem via Trace-Guided Cost Augmentation

Abstract

The Asymmetric Traveling Salesman Problem (ATSP) is one of the most fundamental and notoriously challenging problems in combinatorial optimization. We propose a novel continuous relaxation framework for ATSP that leverages differentiable constraints to encourage acyclic structures and valid permutations. Our approach integrates a differentiable trace-based Directed Acyclic Graph (DAG) constraint with a doubly stochastic matrix relaxation of the assignment problem, enabling gradient-based optimization over soft permutations. We further develop a projected exponentiated-gradient method with adaptive step size to minimize tour cost while satisfying the relaxed constraints. To recover high-quality discrete tours, we introduce a greedy post-processing procedure that iteratively eliminates subtours through cost-aware cycle merging. Empirically, our method achieves state-of-the-art performance on standard asymmetric TSP benchmarks and demonstrates strong scalability and accuracy, particularly on large or highly asymmetric instances where heuristic solvers such as LKH-3 often struggle.

1 Introduction

The Asymmetric Traveling Salesman Problem (ATSP) is one of the most fundamental and notoriously challenging problems in combinatorial optimization. Many important combinatorial optimization (CO) problems are theoretically reducible to ATSP. Recent work by Pan et al. [22] has shown that such reductions are not only of theoretical interest but can also be practically feasible for certain small-scale CO tasks, suggesting that advances in ATSP solvers could yield broad benefits across the CO landscape.

Although the symmetric TSP is a special case of ATSP, solvers designed specifically for the asymmetric case often perform poorly on symmetric instances. Conversely, any ATSP instance can be transformed into a symmetric TSP by doubling the number of nodes [9], enabling symmetric solvers to be applied to asymmetric problems. Consequently, much of the CO community has focused on the symmetric TSP [1, 8, 20, 26], leading to the development of increasingly effective handcrafted and learning-based heuristics. Recently, methods such as those proposed by Drakulic et al. [4, 5], Pan et al. [22] have employed reinforcement learning to directly search for optimal ATSP tours. These approaches provide faster inference compared to state-of-the-art heuristics such as LKH-3 [8], but typ-

ically yield lower-quality solutions-even though LKH-3 was originally developed for the symmetric TSP.

Assignment-problem (AP) relaxations have long been used to provide strong lower bounds within branch-and-bound and branch-and-cut TSP solvers [17, 19], and they remain effective for large-scale asymmetric instances. Several constructive heuristics, including the Karp-Steele Patching (KSP) algorithm [12, 13] and its variants, attempt to lift assignment solutions into valid tours. However, based on existing open-source implementations, these heuristics often underperform relative to LKH-3. Interestingly, AP relaxations tend to be tighter for asymmetric TSPs than for symmetric ones, motivating the development of stronger heuristics grounded in AP formulations.

A general TSP can be formulated as a linear assignment problem with an additional constraint ensuring that the solution forms exactly one cycle [9]. We observe that this constraint is equivalent to requiring all proper subgraphs of the assignment to be Directed Acyclic Graphs (DAGs). Inspired by recent progress in differentiable DAG learning [25, 28, 29], we propose a novel continuous relaxation of the TSP that incorporates DAG constraints into the assignment formulation. We show that the global minimum of this relaxed problem exactly recovers the true TSP solution.

Although the proposed relaxation is non-convex, we find empirically that local minima obtained via approximate projected gradient descent often yield high-quality solutions. In many cases, our method recovers valid tours with shorter lengths than those produced by LKH-3. For instances where the relaxed optimization does not directly yield a valid tour, we introduce a simple greedy post-processing heuristic to eliminate subtours and construct feasible solutions. Finally, we apply a 2-opt local refinement to further improve tour quality. Together, these components constitute a new heuristic for solving general (asymmetric, non-metric) TSP instances that consistently outperforms LKH-3, particularly on large-scale problems. When applied to small-scale symmetric TSPs, our method remains competitive-achieving slightly lower solution quality than LKH-3 but offering significantly faster runtime performance.

In summary, our main contributions are:

Novel TSP formulation: A novel exact continuous formulation that suppresses subtours via a trace-based acyclicity loss.

Soft assignment optimization: A doubly stochastic matrix relaxation of ATSP solved via projected exponentiated gradient descent.

Efficient greedy decoding: A post-processing algorithm that reconstructs valid tours from fractional assignments with minimal additional cost.

Strong empirical performance: Extensive experiments show our method outperforms LKH-3 on ATSP benchmarks and scales well to large problem sizes.

2 Related Works

Learning-based TSP Solvers Recently, learning-based approaches have been increasingly applied to combinatorial optimization (CO) problems, including the Traveling Salesman Problem (TSP). While many existing works focus on the Euclidean TSP, where inputs are given as point coordinates [11, 14, 15, 18, 20, 23, 24, 31], more recent studies [4, 5, 16, 22] have extended these methods to handle general asymmetric TSPs. Kwon et al. [16] treat distance matrices directly as transformer input tokens, which is computationally inefficient and unsuitable for large graphs. Subsequent methods [4, 5, 22] address this limitation by encoding distance matrices more efficiently, enabling scalability to graphs with thousands of nodes. These approaches [4, 5, 16, 22] generate tours in an auto-regressive manner, formulating the tour-construction process as a Markov Decision Process (MDP). Reinforcement learning is then applied to optimize over the MDP, thereby avoiding reliance on expensive ground-truth optimal solutions.

In contrast to these auto-regressive frameworks, Min et al. [20] proposed a surrogate loss based on assignment-problem relaxations of the TSP. However, their approach is limited to the Euclidean setting and only enforces the absence of length-2 loops in the solution, necessitating a post-processing step to construct valid tours.

Learning-based solvers generally outperform classical heuristics such as LKH-3 in terms of computational efficiency, although they may produce slightly less accurate tours. In reinforcement learning

settings, high-quality solutions from fast heuristics (even if suboptimal) can serve as valuable offline data for off-policy algorithms, potentially accelerating training and improving solution quality.

Beyond methods that directly generate tours from node coordinates or distance matrices, another line of research [27, 30] focuses on learning improved search strategies-such as branch-and-bound or local search policies. These approaches replace handcrafted heuristics in traditional solvers with learned decision mechanisms, leading to more adaptive and data-driven optimization behavior.

Assignment Problem Relaxations for TSP Relaxations of the Traveling Salesman Problem (TSP) based on the classical linear Assignment Problem (AP) have been extensively studied due to their computational efficiency and strong theoretical foundations [9, 17]. These relaxations typically involve solving a linear program in which the tour constraints are relaxed, allowing the solution to contain multiple disconnected subtours. To recover a valid TSP tour, additional strategies such as iterative subtour elimination or branch-and-bound frameworks are employed [19]. Although AP relaxations do not always yield feasible tours directly, they provide tight lower bounds [6] and serve as key components in exact solvers and heuristic algorithms. Several tour-construction heuristics have also been developed based on AP relaxations [12, 13]; however, existing open-source implementations often perform worse than modern LKH-3 solvers. For a more comprehensive discussion, we refer readers to Cirasella et al. [2], Glover et al. [7], and Öncan et al. [21].

3 Exact Continuous Relaxation of TSP

Let D ∈ R n × n + be the non-negative distance (or cost) matrix, where its entry d ij denotes the distance or cost of traveling from node i to node j . The general TSP seeks a tour (i.e., a permutation of the nodes forming a single cycle that visits each node exactly once) that minimizes the total travel cost. This can be formulated as:

X forms a single Hamiltonian cycle (no subtours) .

Here, X is a permutation matrix corresponding to a single cycle over the nodes. The final constraint prohibits subtours-i.e., disjoint cycles that do not span all nodes-ensuring that the solution corresponds to a valid TSP tour.

Differentiable DAG Constraints Recently, a family of differentiable constraints for enforcing acyclicity in graphs was proposed in the literature [25, 28, 29]. For any positive adjacency matrix X , acyclicity can be enforced by the following constraint:

In this constraint, each term tr( X i ) is strictly positive when X contains cycles of length i , since all entries of X are positive. Therefore, setting the total sum to zero guarantees that there are no cycles of any length, ensuring that X represents a Directed Acyclic Graph (DAG).

Based on these differentiable DAG constraints, we propose a novel continuous relaxation of the TSP by enforcing that any proper subgraph of the assignment matrix X forms a DAG. This leads to the following formulation:

where D n = { X ∈ [0 , 1] n × n ∣ ∣ X1 = 1 , 1 ⊤ X = 1 ⊤ } is the set of doubly stochastic matrices of size n × n .

One appealing property of this relaxation is its tightness, as stated below:

Proposition 1 (Exactness of the Continuous Relaxation) . The global minimum of the relaxed problem in (3) coincides with the optimal solution of the original TSP .

Proof. To prove the result, it suffices to show that any fractional matrix X ∈ D n cannot satisfy the constraint ∑ n -1 i =1 tr( X i ) = 0 . In other words, all feasible solutions must be permutation matrices representing Hamiltonian cycles.

The proof follows from the Birkhoff-von Neumann theorem [10], which states that any doubly stochastic matrix X ∈ D n can be expressed as a convex combination of permutation matrices:

where P i are permutation matrices.

Assume for contradiction that a fractional X ∈ D n satisfies the constraint in (3). Then its Birkhoff-von Neumann decomposition contains at least two distinct permutation matrices P i . If any P i corresponds to a subtour (i.e., not a single Hamiltonian cycle), then tr( P j i ) > 0 for some j < n , and hence tr( X j ) > 0 for some j < n due to linearity and positivity. This violates the constraint ∑ n -1 j =1 tr( X j ) = 0 , making X infeasible.

Even if all P i represent valid Hamiltonian cycles, a convex combination of two or more distinct tours will produce fractional entries in X . These fractional values imply the presence of edges from multiple tours, which results the presence of subtours and nonzero traces in X j for some j < n . Thus, X would again be infeasible.

Therefore, the only feasible solutions are permutation matrices representing single Hamiltonian cycles. Among them, the one minimizing ⟨ D , X ⟩ is exactly the TSP solution.

4 Trace Guided Tour Search

To efficiently optimize the relaxation in Equation (3), we introduce an auxiliary variable Y ∈ R n × n with elementwise constraints Y ⩽ 0 , and define the assignment matrix as X = exp( Y ) , where the exponential is applied elementwise. This reparameterization ensures that X is strictly positive and facilitates optimization over the interior of the doubly stochastic polytope D n . Compared to optimizing directly over X , working in the log-domain (i.e., optimizing Y ) avoids boundary issues, improves numerical stability, and enables smooth gradients that are more amenable to gradient-based learning methods. The relaxed optimization problem is then formulated as:

where D is the cost matrix, and the trace constraint eliminates self-loops and short cycles. By introducing a Lagrange multiplier λ , the constrained problem can be rewritten as the following unconstrained objective:

The gradient of this objective with respect to Y is given by:

where ⊙ denotes elementwise multiplication. This formulation enables efficient and scalable optimization while preserving the structural constraints required by the TSP.

̸

Approximate Projected Gradient Descent for Exponentiated TSP Relaxation: To optimize the exponentiated relaxation of the TSP problem, we propose an approximate projected gradient descent algorithm over the log-domain variable Y . The algorithm initializes Y as the negative of the cost matrix, Y ←-D , and iteratively updates it using the gradient of the relaxed objective. At each step, the gradient is computed and used to update Y with a step size α . To maintain feasibility within the doubly stochastic set, we apply Sinkhorn [3] normalization to the exponentiated matrix exp( Y ) and then project back to the log-domain via log( · ) , ensuring that exp( Y ) ∈ D n . The step size α is dynamically adjusted based on whether the solution to the linear assignment problem (using -Y as the cost) changes after the update: if it does, the step size is reduced; otherwise, it is increased to accelerate convergence. This adaptive strategy helps balance exploration and stability. The process terminates either after a fixed number of iterations or once a valid TSP tour (a single loop) is found in the assignment. The algorithm returns the optimized Y and the corresponding assignment matrix exp( Y ) .

Approximate Gradient Computation: In our gradient expression, the term [ ∑ n -2 i =0 ( i +1)exp( Y ) i ] ⊤ involves computing a weighted sum of matrix powers, which can be computationally expensive. Although efficient approximations such as the truncated Neumann series proposed in Zhang et al. [28] can reduce the number of operations to O (log n ) matrix multiplications, the cost remains significant for large instances due to the O ( n 3 ) complexity of matrix multiplication. To address this, we adopt a lightweight approximation by substituting exp( Y ) with the hard assignment matrix corresponding to the current solution π prev . Since permutation matrices are sparse and binary, their powers are simply cyclic shifts, and the sum can be computed in O ( n 2 ) time. This approximation significantly accelerates training while preserving useful structural signals from the current assignment during optimization.

Greedy Tour Decoding from Invalid Assignments: Although the proposed optimization approach typically converges rapidly for asymmetric TSP instances, there exist hard cases where convergence to a valid permutation (i.e., a single-loop tour) does not occur. To address such situations, we apply a greedy decoding strategy to extract a valid TSP tour from the relaxed assignment matrix. Specifically, we begin by selecting a random starting node and iteratively follow the highest-assigned edge (i.e., the node with the largest assignment probability) to determine the next node in the tour. If the selected

edge would result in the formation of a premature loop or revisiting a node, we instead choose the closest unvisited node according to the original cost matrix. This process continues until all nodes have been visited exactly once, ensuring the construction of a valid Hamiltonian cycle. This decoding heuristic is simple, efficient, and empirically effective in recovering feasible tours from imperfect or fractional assignments.

Time Complexity. The dominant computational cost in each iteration of the proposed algorithm lies in solving the linear assignment problem for approximating exp( Y ) , which has a worst-case time complexity of O ( n 3 ) when using the Hungarian algorithm. However, in practice, the Hungarian algorithm is typically much faster due to early termination and efficient implementations. Additionally, viewing the assignment step through the lens of primal-dual linear programming reveals an opportunity to reuse the dual variables as warm starts for subsequent iterations, potentially accelerating convergence. Alternatively, the assignment problem can also be solved using the Auction algorithm, which offers better parallelism and is more amenable to GPU implementation. This opens up promising directions for future work, where the entire optimization process could be ported to GPUs to achieve significant speed-ups and scalability for large-scale TSP instances.

Local Search Refinement. To further improve the quality of the obtained tour, our implementation applies a simple 2-opt heuristic as a post-processing step. We observe that for asymmetric TSP instances, the 2-opt heuristic often brings negligible improvements due to the directional nature of the cost structure. In contrast, for symmetric TSPs, the 2-opt refinement can lead to noticeable performance gains by eliminating crossing edges and reducing the overall tour length. While 2-opt is lightweight and easy to implement, it may be suboptimal for larger or more complex instances. As a future direction, one could consider replacing the 2-opt heuristic with more powerful local search methods such as the Lin-Kernighan-Helsgaun (LKH) algorithm [8]. In such a setup, our learned solution could serve as a high-quality initialization for LKH, potentially reducing the number of local search iterations required and improving overall efficiency.

5 Experimental Results

Hardware All experiments were conducted on a workstation equipped with an AMD Ryzen 9 3900X CPU, 128GB DDR4 memory, and an NVIDIA RTX 3090 GPU. The LKH-3 heuristic was implemented in C++ and executed in a single-threaded mode on the CPU. Our proposed method was implemented using Python and SciPy, and it was also executed on a single CPU process without utilizing the GPU. In contrast, all learning-based approaches (e.g., GOAL and MatPOENet) were executed on the GPU to leverage hardware acceleration during inference.

Dataset We evaluate our method on randomly generated Asymmetric TSP (ATSP) instances of sizes 20, 50, 100, 200, 500, and 1000. Following prior works [16], the cost matrices for these instances are generated by sampling each entry independently from a uniform distribution over the interval [0 , 1] . Kwon et al. [16] add an post-processing procedure to ensure the triangle inequality in enforced in the generated ATSP, while we also consider the cases where the triangle inequality is not satisfied. For solvers that require integer-valued cost matrices, we scale the costs by 10 6 and round them to the nearest integer to preserve precision. Additionally, we evaluate our approach on symmetric TSP instances of sizes 20 and 50. In line with previous settings, we first sample n random points in 2D space from a uniform distribution over [0 , 1] 2 , and use the Euclidean distances between these points to construct the symmetric cost matrices.

Competitors We compare our method against several strong baselines, including both classical heuristics and recent learning-based approaches. Specifically, we consider LKH-3 , a state-of-theart heuristic algorithm for solving TSP, GOAL [5], a reinforcement learning-based approach using graph neural networks, and MatPOENet [22], a transformer-based model for solving combinatorial optimization problems. For GOAL and MatPOENet, we use their publicly released pretrained weights to ensure a fair comparison. For LKH-3, we evaluate three configurations: (i) one run with 500 search trials, (ii) one run with 10,000 search trials, and (iii) ten runs with 10,000 search trials each. These configurations allow us to assess both the performance and robustness of LKH-3 under varying computational budgets. Exact solvers such as Gurobi and Concorde [1] are not compared as they are often too slow for large scale problems.

5.1 Asymmetric Problems

Small Scale Problems The comparison of running time and solution quality for small-scale problems is presented in Table 1 and Table 2. Our method, while yielding slightly worse average performance than LKH-3, demonstrates significantly faster running times. For the learning-based approaches, which are trained on ATSP witht triangle inequality instances, their performance degrades substantially on purely random ATSP problems-underperforming all other competitors. This indicates that current neural combinatorial solvers may struggle to generalize across different problem distributions. Additionally, MatPOENet performs considerably worse on our test set than reported in its original paper, suggesting a potential distribution shift between the two test scenarios.

Surprisingly, GOAL achieves the best performance on asymmetric TSP20 with triangle inequality, outperforming the state-of-the-art LKH-3 heuristic by a substantial margin. Given that GOAL was

trained on 100-node triangle asymmetric instances, this result suggests that training neural solvers on large, structured, and challenging problems may enhance their generalization to smaller problem instances.

Medium and Large Scale Problems The comparison of running time and solution quality for medium- and large-scale problems is presented in Table 3, Table 4, Table 5, and Table 6. Consistent with the small-scale setting, learning-based approaches perform significantly worse than others on purely random asymmetric TSPs. In these settings, our algorithm outperforms LKH-3(1,500) for graphs with more than 100 nodes, and even surpasses LKH-3(1,10000) for graphs with more than 200 nodes. As the graph size increases, our method increasingly outperforms all LKH-3 heuristics, with the performance gap becoming more pronounced for larger instances. On 1000-node problems, our algorithm achieves over 20% improvement compared to LKH-3(10,10000) and over 70% compared to LKH-3(1,500). Meanwhile, our runtime on large graphs is comparable to that of LKH-3(1,500) and remains faster than the learning-based approaches. On asymmetric TSPs with triangle inequality, similar trends are observed. Our method continues to outperform LKH-3 on large graphs, although the performance margin is smaller compared to the purely random setting.

In terms of running time, it is notable that, despite our implementation not being carefully optimized, its execution speed is often faster than that of LKH-3. On very large graphs, our method can outperform learning-based approaches, even when those methods are accelerated by GPUs. Furthermore, most operations in our algorithm are amenable to parallel execution and could potentially benefit significantly from GPU acceleration.

5.2 Symmetric Problems

Although our algorithm is not specifically designed for symmetric TSPs and does not exploit any symmetry properties, we evaluated its performance on small-scale symmetric instances. The results are presented in Table 7. The performance trends closely resemble those observed on triangle asymmetric TSPs.

6 Conclusions and Future Works

In this work, we introduced a novel continuous relaxation framework for the Asymmetric Traveling Salesman Problem (ATSP) that combines differentiable acyclicity constraints with doubly stochastic matrix relaxations. Our key innovation lies in the formulation of ATSP as a constrained optimization problem where trace-based DAG constraints naturally eliminate subtours while maintaining differentiability for gradient-based optimization. The proposed projected exponentiated gradient method with adaptive step sizes demonstrates efficient convergence properties, and our greedy post-processing procedure effectively recovers valid tours from fractional assignments.

Through extensive experiments on standard benchmarks, we have shown that our method achieves state-of-the-art performance on asymmetric TSP instances, particularly excelling on large-scale problems where traditional heuristics like LKH-3 struggle. The results reveal several important insights: (1) our continuous relaxation provides a tight formulation that frequently converges to valid tours without post-processing, (2) the approach scales favorably with problem size, maintaining both computational efficiency and solution quality, and (3) the method remains competitive even on symmetric TSP instances despite not being specifically designed for them.

The success of our framework suggests several promising directions for future research. First, the differentiable nature of our formulation makes it potentially amenable to integration with learningbased approaches, where the solver could be trained to predict good initializations or adaptive penalty parameters. Second, the trace-based constraints may generalize to other combinatorial optimization problems that require acyclic substructures. Finally, investigating more sophisticated optimization techniques within our framework could lead to further improvements in both solution quality and computational efficiency.

This work bridges an important gap between continuous relaxations and discrete combinatorial optimization, offering a new perspective on solving challenging problems like ATSP through differentiable programming. The demonstrated effectiveness of our approach on large-scale instances suggests that similar techniques could prove valuable for other NP-hard optimization problems where traditional methods face scalability limitations.

Acknowledgments and Disclosure of Funding

This research is supported by Responsible AI Research Center (RAIR) and the Ministry of Education, Singapore, under its Academic Research Fund Tier 1 (A-8001814-00-00).

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NeurIPS Paper Checklist

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Appendix: Solving Asymmetric Traveling Salesman Problem via Trace-Guided Cost Augmentation

A Linear Assignment Problem

The linear assignment problem aims to determine a permutation matrix that minimizes the total assignment cost given by a cost matrix D ∈ R n × n , whose entries are denoted by d ij . Let X ∈ R n × n be the assignment matrix, where x ij = 1 indicates that row i is assigned to column j , and zero otherwise. The problem can be formulated as the following linear program:

The dual problem introduces variables u i and v j corresponding to the row and column constraints, respectively, and is given by:

Strong duality holds for this problem, guaranteeing that the optimal values of the primal and dual problems are equal when a feasible solution exists.

The Hungarian Algorithm is, in fact, a primal-dual method for solving the Linear Assignment Problem (LAP). This insight allows us to exploit the structure of Algorithm 1, where multiple LAPs are solved iteratively. Specifically, we can retain the previously optimal dual variables u and v and use them as the initialization for subsequent runs of the Hungarian Algorithm. This warm-start strategy significantly accelerates convergence, as the optimal dual variables often remain close across successive LAP instances.

B Further Refining Performance by Branch-and-Bound

We further refined our implementation by incorporating two key strategies: (1) reusing the dual variables u and v , as described in the previous section, and (2) introducing a simple branch-and-bound heuristic. In our branch-and-bound strategy, we first solve the Linear Assignment Problem (LAP) to obtain a lower bound for the ATSP instance. We then identify the edge with the highest cost in the LAP solution and branch on it. For each branch, we apply Algorithm 1 to compute a potentially global upper bound, while the LAP provides a local lower bound. Our heuristic allows up to five branch-and-bound iterations.

With these enhancements, the results on large-scale ATSP instances satisfying the triangle inequality are presented in Table 8. As shown, reusing the dual variables accelerates our algorithm by up to a factor of five compared to the original implementation. Additionally, the branch-and-bound strategy offers further performance improvements. Similar gains are observed on purely random ATSP instances, as shown in Table 9.

Table 9: Comparison of methods on large-scale pure random ATSP problems (TSP500 and TSP1000). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. Our approaches outperforms the state-of-the-arts LKH-3 heuristics in terms of solution quality. The performance of learning based approaches is far worse than their reported results as they are trained on ATSP with triangle inequality and they failed to generalize to random ATSP.

C Experimental Results on TSPLIB instances

We evaluated our algorithm on large-scale ATSP benchmark instances from TSPLIB, particularly those derived from stacker crane problems. Our method consistently outperforms LKH-3 (1-100) in terms of solution quality while also requiring less computational time. These results demonstrate the practical effectiveness of our approach on real-world structured benchmarks, further validating its scalability and competitiveness beyond synthetic instances.

D Comparison with ILP Solver

We conducted a preliminary comparison between our method and an exact ILP solver on both symmetric and asymmetric TSP instances. Specifically, we tested the solver on 100-node symmetric problems and 500-node asymmetric problems and found that the ILP solver was able to return optimal solutions within a reasonable time budget.

Due to time constraints, we evaluated both methods on 10 randomly sampled instances for each case. The results are summarized below. While the ILP solver guarantees optimality, our method achieves competitive solution quality with significantly faster runtimes, particularly for larger asymmetric instances-highlighting its scalability and practical utility for large-scale problems.

D.1 Performance comparison on 100-node symmetric TSP problems

D.2 Performance comparison on 500-node asymmetric TSP problems

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper propose to continuously relax the ATSP of discrete variables, and uses differentiable directed acyclic graph techniques and assignment problem relaxation to achieve gradient-based optimization. Since the relaxation is non-convex, this paper develops an adaptive step-size projected gradient descent to find high-quality solutions; for cases where the relaxation optimization does not yield a valid path, a simple greedy post-processing heuristic method is introduced to eliminate sub-cycles and construct feasible solutions.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- The proposed method is reasonable and sound.\n- Extensive experiments show that the proposed method generally outperforms LKH-3 on the ATSP benchmark and has good generalization ability for problem sizes.\n\n**Weaknesses**\n- It is unclear how many chances there are to find a valid solution directly and find a valid solution via continuous iteration, respectively.\n- No ablation experiments are included to analyze the effectiveness of the proposed method components.\n- The authors did not report the results without 2-opt.\n- The article mentions that Gurobi and Concorde are not included as baselines because they run very slowly on large scales. What about small-scale problems?\n\nQUESTIONS:\n- What are the results on ATSPLIB [1] and real-world ATSP datasets [2]?\n\n- Table 7 is about a small-scale symmetric TSP problem. How does it perform on a larger-scale symmetric TSP problem?\n\n- Why is the proposed algorithm not as good as GOAL on ATSP20 (Table 2)?\n\n- Why does the proposed algorithm perform worse than LKH3 on a small scale but better on a large scale (Table 1)?\n\n- The method in this paper does not involve learning. Why mention learning-based methods separately in related work?\n\n- Does the initial solution have a big impact on the solution quality?\n\n- The headers of Tables 1 to 6 should be ATSP instead of TSP. Please do not omit it. This will confuse people at first glance, especially since Table 7 is about TSP.\n- Also see weaknesses.\n\n[1] http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/atsp/\n\n[2] Neural Combinatorial Optimization for Real-World Routing, arxiv 2025.\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

comments: SUMMARY:
This paper presents an innovative continuous relaxation framework for the Asymmetric Traveling Salesman Problem (ATSP). The core idea is to combine a differentiable, trace-based Directed Acyclic Graph (DAG) constraint with a doubly stochastic matrix relaxation to formulate the problem. This approach effectively suppresses subtours, enabling the use of gradient-based optimization to find solutions. The proposed method demonstrates superior performance and scalability compared to state-of-the-art methods like LKH-3, especially on large-scale ATSP benchmarks.

STRENGTHS AND WEAKNESSES:
**Strengths:**
* The paper proposes a novel, differentiable, and exact continuous formulation for ATSP.
* On large-scale ATSP instances (with more than 200 nodes), the method's solution quality and runtime are superior to the highly-optimized LKH-3 heuristic.
* The algorithm shows convincing performance with solid evaluation.

**Weaknesses:**
* The experiments report average and median results  but do not provide error bars or confidence intervals.
* A valuable direction for future work would be to explore the algorithm's performance in worst-case scenarios.

QUESTIONS:
Please refer to weakness.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present a local search algorithm for finding solutions to the TSP. The algorithm is based upon incorporating a differentiable formulation of the constraints into the objective and subsequently applying gradient descent to find solutions. Additionally, the resulting TSP formulation is relaxed to allow for continuous solutions. Improvements and details regarding runtime and numerical stability are discussed, as well as an approach to recover feasible solutions from invalid assignments. Evaluation is performed on randomly generated TSP instances, and the presented approach is compared to SotA local search heuristics and learning-based approaches.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- The key concepts of the paper are laid out clearly and in an easy-to-follow manner.\n- The algorithm shows convincing performance on the evaluated random instances with respect to solution quality and needed runtime in comparison to state-of-the-art methods, especially on large instances.\n\n**Weaknesses**\n- The evaluation is restricted to purely randomly generated TSP instances.\n- The paper has several grammar and spelling mistakes, e.g.:\n    - line 71 \"auto-aggressive way\"\n    - line 72 \"learning approach are applied\"\n    - line 82 \"Besides on those approaches\"\n    - line 83 \"choose learn better\"\n\nQUESTIONS:\n1. How is the multiplier $\\lambda$ chosen? Which impact does the choice have on the performance of the algorithm?\n2. How dependent is the output on the chosen initialization $Y = -D$? Would the algorithm benefit from re-running using different initializations?\n3. How does the presented algorithm perform on benchmark instances from i.e. TSPLIB?\n4. How does the approximate gradient computation affect the output of the algorithm? Is the purpose of the approximation purely to improve runtime or does it also improve the final output of the algorithm?\n5. On line 160 it is stated that stepwidth $\\alpha$ is increased to accelerate convergence. \nUsually in gradient descent, one increases the stepwidth to achieve greater exploration, while decreasing it leads to convergence to a local minimum. Why does increasing the stepwidth accelerate convergence for this algorithm?\nFurthermore, it is stated that the adaptive strategy helps to balance exploration and stability. It is not entirely clear what is meant by stability here. Does stability here mean the ability of the algorithm to converge to a local solution?\n6. Proposition 1 states that any fractional matrix cannot satisfy the constraint. Does this not imply that the feasible solutions to the relaxed problem defined in Eq. (3) and of the original problem definition in Eq. (1) are the same, and therefore (3) is not a relaxation of (1), but (3) is an equivalent way of stating (1)?\n7. Additionally, on the small instances (TSP20 and TSP50) a comparison to the optimal solution found by an ILP solver would be interesting. This could also be useful in validating the experimental results as e.g. in Table 2 multiple algorithms consistently find the same solution (indicated by the avg. and median gap both being 0.0%), which can be a sign that they all have found the optimal solution.\nHowever, there exists an algorithm that consistently finds better solutions (GOAL in this case). Comparing to the optimal solution found by an ILP solver could be used to verify that no error has been made when taking these measurements.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new continuous relaxation framework for the ATSP using trace-based acyclicity constraints and doubly stochastic matrix optimization. The method integrates differentiable DAG constraints to eliminate subtours, employs projected exponentiated gradient descent for optimization, and uses a greedy post-processing step to recover valid tours. Experiments demonstrate state-of-the-art performance on large-scale ATSP instances, outperforming LKH-3 and learning-based baselines.\n\nSTRENGTHS AND WEAKNESSES:\nI have to say that I am usually working on the approximation algorithms for the network design problem (etc., approximating ATSP). By reading the abstract, I thought this work was related to my field. But, it turns out that it goes beyond my experience after reading the whole paper. I am not the right person to judge the contribution and novelty of the paper. \n\nFrom my perspective, the paper has the following strengths:\n\n1. The idea of using trace-based constraints looks interesting to me. This approach is to enforce acyclicity in ATSP relaxations, and Proposition 1 establishes that the global minimum of the relaxed problem recovers the exact TSP solution; this is also an interesting idea to me.\n\n2. From my view, the gradient approximation (using hard assignments) efficiently addresses the high cost of matrix power computations. However, I don't know whether it's computationally practical (e.g., easy to implement) or not.\n\n3. This paper gives competitive results on symmetric TSPs (as shown in Table 7), suggesting potential generality beyond ATSP.\n\nThe main weakness that I can see is that:\n\nThe non-convex optimization (Equation 6) lacks a theoretical analysis of convergence rates or approximation bounds. The adaptive step size heuristic is pragmatic but ad hoc. This probably can become a big problem in other datasets.\n\nQUESTIONS:\nI don't have specific questions.\n\nLIMITATIONS:\nn/a", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}