Title: Differentiable extensions with rounding guarantees for combinatorial optimization over permutations
Venue: NeurIPS 2025
Accepted: yes
Keywords: Combinatorial optimization

Permutations

Birkhoff-von-Neumann decomposition

matching

quadratic assignment
Abstract: Continuously extending combinatorial optimization objectives is a powerful technique commonly applied to the optimization of set functions. However, few such methods exist for extending functions on permutations, despite the fact that many combinatorial optimization problems, such as the quadratic assignment problem (QAP) and the traveling salesperson problem (TSP), are inherently optimization over permutations. We present Birkhoff Extension (BE), an almost-everywhere-differentiable continuous polytime-computable extension of any real-valued function on permutations to doubly stochastic matrices. Key to this construction is our introduction of a continuous variant of the well-known Birkhoff decomposition. 
Our extension has several nice properties making it appealing for optimization problems. 
First, BE provides a rounding guarantee, namely any solution to the extension can be efficiently rounded to a permutation without increasing the function value. Furthermore, an approximate solution in the relaxed case will give rise to an approximate solution in the space of permutations. 
Second, using BE, any real-valued optimization objective on permutations can be extended to an almost-everywhere-differentiable objective function over the space of doubly stochastic matrices. 
This makes our BE amenable to not only gradient-descent based optimization, but also unsupervised neural combinatorial optimization where training often requires a differentiable loss. 
Third, based on the above properties, we present a simple optimization procedure which can be readily combined with existing optimization approaches to offer local improvements (i.e., the quality of the final solution is no worse than the initial solution). 
Finally, we also adapt our extension to optimization problems over a class of trees, such as Steiner tree and optimization-based hierarchical clustering.
We present experimental results to verify our theoretical results on several combinatorial optimization problems related to permutations.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors tackle optimization problems over the space of permutations.\n\nThey introduce **Birkhoff Extension (BE)**, a continuous, almost-everywhere differentiable relaxation of permutation functions defined on the Birkhoff polytope (the convex hull of permutation matrices).\n\nThey propose a scoring-matrix-based method for rounding.\n\nThey show good theoretical properties about rounding.\n\nThey also design optimization techniques, e.g., Frank-Wolfe for preserving doubly-stochasticness, and updating the scoring matrix for escaping from local minima.\n\nOn benchmark instances of QAP (from QAPLIB), random-Euclidean TSP, and Directed Feedback Arc Set (on random graphs), BE matches or outperforms Gurobi, classical heuristics, and existing approximation algorithms.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n\n- **[S1] Important Problem**: Optimization problems on permutations are very important with both theoretical and empirical values.\n- **[S2] Novel Differentiable Birkhoff Decomposition:** Although the classical Birkhoff Decomposition has been well known for a long time, the proposed differentiable Birkhoff decomposition based on the scoring matrix is novel and interesting.\n- **[S3] Experiments with Broad Applicability: The proposed framework works across a variety of problems.**\n\n# Weaknesses\n\n- **[W1] High Time Complexity**: O(n^5) time complexity can be prohibitive for large-scale problems. Also, Frank-Wolfe and dynamic score bring additional overheads.\n- **[W2] Probably Not-Very-Practically-Meaningful Theory:**\n    - See [Q1] and [Q2] below\n- **[W3] Missing References:**\n    - Min et al. \"Unsupervised learning for solving the travelling salesman problem.\" NeurIPS’23\n        - This work also discusses unsupervised learning for TSP\n    - Bu et al. \"Tackling prevalent conditions in unsupervised combinatorial optimization: Cardinality, minimum, covering, and more.\" ICML’24\n        - This work also discusses theoretical properties of objective extension and rounding in unsupervised combinatorial optimization, as well as continuous relaxation for problems on trees\n\nQUESTIONS:\n- [Q1] Property 4-2 tells that the rounding result cannot be worse than the scoring matrix, but I am wondering whether the rounding can actually get better than the scoring matrix when S is “sufficiently close to some permutation P”.\n- [Q2] Theorem 2.11 only tells us the existence of such an S, but I am wondering whether we can have any results on, e.g., how easily we can find such an S.\n\nLIMITATIONS:\nYes. Limitations are discussed in Section 4, “Concluding Remarks”.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 4, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents Birkhoff Extension (BE), an almost-everywhere differentiable continuous extension that lifts any function defined on permutation matrices to the Birkhoff polytope. The key idea is a score-induced Birkhoff decomposition, which selects permutations in a fixed order determined by a score matrix. The proposed method exhibits several appealing properties, such as minima correspondence and a rounding guarantee. The authors further develop a Frank-Wolfe optimizer with optional dynamic score updates to help escape local minima, and they apply this framework to unsupervised neural combinatorial optimization. Experimental results on QAP, TSP, and DFASP tasks demonstrate the effectiveness of the proposed approach.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n1. The score-induced Birkhoff decomposition is a novel construction that selects a permutation order based on a score matrix, effectively resolving the non-uniqueness of the decomposition. This design enables differentiable extensions of discrete objectives and facilitates gradient-based optimization over permutations.\n\n2. The theoretical analysis is solid, covering differentiability, minima correspondence, and a rounding guarantee that ensures relaxed solutions can be safely mapped back to valid permutations without degrading objective quality.\n\n3. The proposed Frank-Wolfe algorithm is general and versatile, applicable to both neural combinatorial optimization and classical matching-based problems. Its effectiveness is empirically validated through comprehensive experiments on QAP, TSP, and DFASP tasks.\n\n## Weaknesses\n1. The decomposition has $O(n^5)$ computational complexity, and while truncated in practice, the paper lacks an analysis of how such truncation affects solution accuracy or theoretical guarantees.\n\n2. The experimental insights are limited. The key hyperparameters, such as truncation depth and score update frequency, are fixed without accompanying ablation studies or sensitivity analysis.\n\n3. The advantage of using the Birkhoff decomposition over direct continuous optimization on the Birkhoff polytope (e.g., using Sinkhorn or projected gradient methods) is not clearly justified, and a more detailed discussion would be beneficial.\n\nQUESTIONS:\n1. How does the number of decomposition terms $k$ affect solution quality? A theoretical bound or empirical curve would be helpful.\n\n2. Can the method scale to larger problems (e.g., $n=1000$) with manageable runtime?\n\n3. Can you provide sensitivity analysis for the key parameters like truncation depth and score update frequency?\n\n4. How does this method compare to direct optimization over the Birkhoff polytope (e.g., Sinkhorn, PGD)?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes an approach for providing an differentiable extension to loss functions on permutation matrices using a novel decomposition of the Birkhoff polytope based on a total ordering of permutations created by the inner product to a particular 'score matrix'. The extension is differentiability almost everywhere since the decomposition is unique. Nonetheless, the extension may not be non-convex and the score matrix can be changed during operation to escape local minima. Additionally the extension has desirable rounding properties such that the permutation matrix obtained from the rounding operation to obtain the best permutation matrix within the decomposition is no worse than the extension's loss. Additionally if the score function is close to permutation than the rounding will only do better.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\nThe paper contributes to combinatorial optimization through the use of differentiable extension. \n\n\nWeakness:\n\nThere a few times things could be made clearer as noted in Questions below. \n\nThe practical implications of the truncation to 5 terms in the Birkhoff decomposition are not addressed by any of the theoretical results. This creates a gap between the theory and practice that deserves more discussion.\n\nQUESTIONS:\nThe definition of $\\mathcal{P}(B_k)$ could be made clearer. I understand it as the set of all permutations that are non-negative components of $B_k$, but note that $B_k$ itself is not double stochastic. Line 219 discusses maximum weight matching problem for the Birkhoff, but the residual $B_k$ isn't in the Birkhoff polytope. This should be clarified. \n\nIn Algorithm 3 and 4,  there could be a discussion of the necessitate of starting from a *dense* random double stochastic matrix $A$ (which should really be denoted $A_1$) compared to either the dense barycenter (a uniform double stochastic matrix) or a sparse combination. Especially since only 5 terms are initially used. Clearly there will be issues if the initial $A_1$ is not dense. But does it need to be random? Especially since the score matrix is is used the barycenter seems a logic initialization. \n\nI assume that automatic gradient are necessary for the Frank-Wolfe algorithm as the relationship between extension and A through $\\alpha_i$ is non-trivial to express. \n\nMinor:\nLine 319 minor typo \"simple constraint\" -> \"simple constraints\"\n\nLIMITATIONS:\nLimitations are mentioned, but more discussion could be made.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a continuous and almost everywhere differentiable extension from functions over permutations to functions over doubly stochastic matrices. The starting point is the Birkhoff decomposition, which allows us to decompose any doubly stochastic matrix into a convex combination of a relatively small number of permutation matrices. If the doubly stochastic matrix $A$ is decomposed as $\\sum_k \\alpha_k P_k$, where the $P_k$ are permutation matrices, we might extend a function $f$ of permutation matrices to doubly stochastic matrices by defining $F(A) = \\sum_k \\alpha_k f(P_k)$. This would indeed be an extension, since when $A$ is a permutation matrix the only valid way to express it as a convex combination of permutation matrix is to put all the weight on $A$ itself. The main challenge with this approach is that the decomposition is not unique, and so depending on how we choose the decomposition to use for each doubly stochastic matrix $A$, the function $F$ may behave erratically. The core idea behind this paper is to introduce a total ordering over permutation matrices and to use that ordering to standardize the decomposition (by preferring to put weight on early permutation matrices in the ordering). The authors argue several nice properties hold with this modification: the extension $F$ is continuous and almost everywhere differentiable. Moreover, they define a collection of total orderings where the decomposition can also be computed in polynomial time (the orderings are those induced by inner products with scoring matrices, and as long as the scoring matrices have a bit of noise, this is guaranteed to be an ordering without ties). In this case, the decomposition of an $n \\times n$ doubly stochastic matrix can be computed on $O(n^5)$ time. This means that the extension can be evaluated in $O(n^5)$ time to obtain the decomposition, and $O(n^2)$ calls of the function $f$ over permutations.\n\nIn addition to the above results, the authors also provide a number of other useful results: a version of the Frank-Wolfe optimizer that can be used to optimize $F$ without needing an expensive projection back onto the set of doubly stochastic matrices, tricks for changing the scoring matrix used to define the ordering to prevent a current optimization iterate from being at a local minima of the extended function, and extensions for functions of rooted binary trees.\n\nFinally, the authors apply their extension to several combinatorial optimization problems (the quadratic assignment problem, traveling salesperson problem, and directed feedback arc set problem). Surprisingly, the extension finds better solutions than Gurobi and heuristic methods for QAP and DFASP under some regimes.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n- The paper is very well written and despite being about a topic that I have relatively little background in, I was able to understand the main claims and arguments of the paper.\n- Deep learning has shown that gradient based optimization can often work well even when we cannot prove that it should (e.g., when run on non-convex functions with many local minima). This paper provides a framework allowing the application of that machinery to combinatorial optimization problems over permutations, and the experiments show some evidence that this approach can in some cases outperform state-of-the-art solvers.\n- The authors go well beyond establishing the basic idea and provide nice ideas for the practical application of their extension (e.g., updating the score matrix used to define the total ordering to escape local minima during optimization).\n\n# Weaknesses\n- The computational cost of computing the extension once is polynomial, but still relatively expensive ($O(n^5)$ time to compute the decomposition and $O(n^2)$ evaluations of the original function on permutations). In the experiments the authors only compute the first $k = 5$ coefficients in the decomposition. I am curious how much this cost can be reduced. I also wonder if there is any way to \"warm-start\" the decomposition so that if $A'$ is close to $A$ (i.e., maybe one step away in the gradient-based optimization), we can leverage the decomposition of $A$ to more efficiently decompose $A'$.\n- It seems like the choice of scoring matrix $S$ might affect some properties of the extension (e.g., is the Lipschitz constant of the extension dependent on the score $S$?). In addition to the paper's results on escaping local minima, it could be interesting to explore whether the choice of $S$ has other impacts that could be optimized for.\n\nQUESTIONS:\nPlease see high level questions in strengths and weaknesses.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}