Title: Graph Alignment via Birkhoff Relaxation
Venue: NeurIPS 2025
Accepted: yes
Keywords: Graph Alignment

Quadratic Assignment Problem

Convex Relaxation

Sensitivity Analysis
Abstract: We consider the graph alignment problem, wherein the objective is to find a vertex correspondence between two graphs that maximizes the edge overlap. The graph alignment problem is an instance of the quadratic assignment problem (QAP), known to be NP-hard in the worst case even to approximately solve. In this paper, we analyze Birkhoff relaxation, a tight convex relaxation of QAP, and present theoretical guarantees on its performance when the inputs follow the Gaussian Wigner Model. More specifically, the weighted adjacency matrices are correlated Gaussian Orthogonal Ensemble with correlation $1/\sqrt{1+\sigma^2}$. Denote the optimal solutions of the QAP and Birkhoff relaxation by $\Pi^\star$ and $X^\star$ respectively. We show that $\|X^\star-\Pi^\star\|_F^2 = o(n)$ when $\sigma = o(n^{-1})$ and $\|X^\star-\Pi^\star\|_F^2 = \Omega(n)$ when $\sigma = \Omega(n^{-0.5})$. Thus, the optimal solution $X^\star$ transitions from a small perturbation of $\Pi^\star$ for small $\sigma$ to being well separated from $\Pi^\star$ as $\sigma$ becomes larger than $n^{-0.5}$. This result allows us to guarantee that simple rounding procedures on $X^\star$ align $1-o(1)$ fraction of vertices correctly whenever $\sigma = o(n^{-1})$. This condition on $\sigma$ to ensure the success of the Birkhoff relaxation is state-of-the-art.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper \"Graph Alignment via Birkhoff Relaxation\" analyzes the Birkhof relaxation of quadratic assignment problems (QAPs) from a theoretical perspective. It considers the setting of inputs following a Gaussian Wigner Model, and gives guarantees based on the dependence of the standard deviation $\\sigma$ of the perturbation (in the input) on the dimensionality of the problem. The analysis results in a state-of-the-art condition for the provable success of the Birkhoff relaxation. It also indicates a phase transition for standard deviations that decay slower. A brief numerical study indicates that the success of the relaxation might go beyond the provable guarantee and that post-processing rounding schemes seem to yield strong results even in the case where the result  of the relaxation itself is provably well-separated from the true permutation.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper addresses an interesting and important problem. To the best of my knowledge, it makes an important theoretical contribution, advancing the provable guarantees in the form of error estimates for the Birkhoff relaxation of the quadratic assignment problem under specific assumptions on the input matrices. Although the numerical results do not necesarily indicate that the error bounds (or conditions) are tight, I consider this work to be an important step towards a better understanding of the Birkhoff relaxation. \n\nIn terms of the presentation, the paper is not very easy to follow. This is partly due to the theoretical, mathematically challenging nature of the topic. Yet, moving some more techincal aspects into the appendix to generate space for a little more context to address a broader audience could be helpful (e.g. possibly introducing the Birkhoff polytope formaly). Some specific aspects that could improve the presentation:\n- I would have found it helpful if alternative relaxation methods were briefly introduced. For instance, I could imagine that \"simplex\" (reference [1]) is a weaker relaxation obtained by dropping the sum-to-one constraint along the columns (or rows), but currently understanding the relaxation requires looking at reference [1]. \n- The variable $J$ (equation after l. 110) is not defined but used many times throughout the manuscript. Its meaning can be deduced from the context but this is not friendly for a quick read. \n- In line 50, I think it should read $B= A+\\sigma Z$ (not $B^{\\pi*}$), and the definition of $B^{\\pi^*}$ follows from $B$ in the next line. \n- After reading in line 123 that the analysis in [1] is restricted to $\\sigma =0$, and that the work presented here allows $\\sigma>0$, I was surprised to read \"we consider a simpler optimization problem corresponding to $\\sigma=0$ in (2)\" in line 157. It subsequently becomes clear that this is just an intermediate step, but informing the reader about the big picture of the proof strategy would be beneficial. \n- The bound $\\sqrt{n}\\geq ||X||_F^2$ is a typo (but it is used correctly, without the square, in the proof). \n- I'd recommend to consider different colors for the middle plot of Fig. 1, or insert a title \"Birkhoff Relaxation for n=400\". As the left plot has the same colors and a legend that does not apply to the middle plot, this can be misleading when having a quick look only. \n- Typo in line 260, \"left plot\" should be \"right plot\".\n\nQUESTIONS:\nPlease consider improving the readability of the manuscript for a broader audience. In particular, please address the explicit suggestions for an improved clarity listed under \"Strength and Weaknesses\".\n\nIn the checklist, the authors state that \"Our code is a simple Python script to solve convex optimization problems using cvxpy. Such an approach is ubiquitous, so we decided not to make the code open-source. However, we are happy to make it open-source if the reviewers deem it to be essential.\"  Despite the simplicity of the experiments, I strongly recommend making the source code publically available. This allows other researchers to reproduce that paper's results exactly with minimal effort.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of graph alignment in the Gaussian Wigner Model. The goal is to align the vertex sets of two correlated matrices A, B where B has an unknown permutation applied to it. The authors consider the Birkhoff relaxation of the problem, which is a well-known (but poorly studied) relaxation of the QAP formulation. When the correlation between A and B is $o(n^{-1})$, the Frobenius norm between the optimal solution and the true permutation matrix is $o(n)$. In turn, this guarantees that a simple rounding technique recovers the true permutation. On the other hand, when the correlation between A and B is $\\Omega(n^{-0.5}),$ then the Frobenius norm between the optimal solution and the true permutation matrix is $\\Omega(n)$. However, the latter result does not rule out the possibility of rounding the relaxed solution.\n\nSTRENGTHS AND WEAKNESSES:\nAnalyzing Birkhoff relaxation has been a difficulty in the graph alignment literature, so I view this paper as making a solid step forward, even though the correlation algorithmic guarantees are weaker than those of competing approaches (GRAMPA, simplex relaxation). Still, empirical results show that the Birkhoff relaxation strategy (+ Hungarian algorithm) performs better than those approaches.\n\nThe cited works are incomplete. Most crucially, reference [8] is cited but not included in Section 1.3. Also, a recent paper of Gaudio, Racz, and Sridhar (“Average-case and smoothed analysis of graph isomorphism”) provides improved guarantees for matching in the noiseless Erdos-Renyi setting. \n\nLine 215: “the triangle’s inequality”\n\nLine 260: “left plot” should be “right plot”\n\nCorollary 2 uses the greedy approach for matching. However, the simulations use the Hungarian algorithm. Therefore, I don’t see the simulations as validating Theorem 1, but rather as a standalone investigation. This shortcoming should be acknowledged, and there should be an empirical evaluation that uses the greedy approach.\n\nQUESTIONS:\nIn Equation (3), don’t we need an upper bound on $\\Vert Z^2 \\Vert_F$ in order to derive a lower bound on $\\Vert I – X^{\\star} \\Vert_F$?\n\nI am not following the equality in Line 198. Could you please clarify?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the graph-alignment (quadratic assignment) problem under the Gaussian Wigner Model and delivers the first guarantees for the Birkhoff relaxation. It shows a sharp phase transition: when noise is $o(1/n)$, the relaxed solution remains close to the true permutation, while for noise on $\\Omega(1/\\sqrt{n})$ it drifts significantly away. The author's analysis relies on a dual certificate that respects the non-negativity constraints of the Birkhoff polytope, combined with sensitivity analysis of convex problems. The experiments are simulated on graphs of up to 500 nodes to verify theoretical results.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n1. The paper provides the first formal guarantee for the Birkhoff relaxation in the graph alignment problems and identifies a meaningful threshold on the noise level that separates success from failure.\n\n2. The paper is well written and easy to follow. The main contributions are presented in a straightforward manner, and the core theoretical result is introduced early on, accompanied by helpful intuition.\n\n## Weaknesses\n1. The relative improvement over prior work is difficult to assess. Although the paper offers a comprehensive literature review, it does not quantitatively compare its theoretical results with existing works. A concise comparison table—summarizing assumptions, noise thresholds, and relaxation types—would help clarify the novelty of this work.\n\n2. The analysis is limited to the correlated Gaussian Wigner model. It is unclear whether the proposed proof techniques can generalize to new settings.\n\n3. While the theory predicts a sharp transition based on the noise level, this phenomenon is not fully explored in the experiments. In particular, it would be valuable to examine how the error evolves more finely across the noise regime, especially when $\\sigma$ lies between $1/n$ and $1/\\sqrt{n}$.\n\nQUESTIONS:\n1. The authors should further discuss the comparison with previous works and the generalizability of theoretical frameworks.\n\n2. Would the authors consider sampling $\\sigma$ more densely within the intermediate range $[1/n,1/\\sqrt{n})]$ and reporting how the empirical error aligns with the predicted behavior?\n\n3. The reported runtime (up to 3 hours) appears significantly higher than related works on graph matching [1, 2], where solving times are typically reported in seconds. Could the authors explain this difference?\n\n4. The authors mention that the code is simple and thus not released. However, reproducibility is critical, especially for optimization problems where solver settings can influence results.\n\n[1]. Bernard, Florian, Daniel Cremers, and Johan Thunberg. \"Sparse quadratic optimisation over the stiefel manifold with application to permutation synchronisation.\" Advances in Neural Information Processing Systems 34 (2021): 25256-25266.\n\n[2]. Dröge, Hannah, et al. \"Kissing to find a match: efficient low-rank permutation representation.\" Advances in Neural Information Processing Systems 36 (2023): 48459-48471.\n\nLIMITATIONS:\nThe authors partially address the limitations of their work in Sections 2 and 6. However, a more explicit discussion of the two main limitations mentioned above would strengthen the paper.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a theoretical analysis of Birkhoff relaxation for graph alignment under the Gaussian Wigner Model (claimed as a first). The authors establish theoretical guarantees showing that when $\\sigma = o(n^{-1})$, the optimal solution $X^* $ approximates the true permutation with $||X^* - \\Pi^*||^2_F = o(n)$, while solutions become well-separated when $\\sigma = \\Omega(n^{-\\frac{1}{2}})$. The work employs dual certificate construction techniques to handle non-negativity constraints and provides rigorous mathematical analysis including eigenvalue separations and concentration inequalities. The paper provides a synthetic simulation showcasing its prowess over the traditional baseline.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n- **Novel theoretical contribution:** A rigorous theoretical analysis of Birkhoff relaxation for graph alignment, which serves as a significant contribution to support the method's empirical success.\n- **Technical rigor:** Mathematical analysis is thorough with dual certification and proofs demonstrate rigorous treatment (eigenvalue separations, concentration inequalities, etc).\n- **Clear phase transition:** Establishes a meaningful theoretical phase regime where alignment succeeds versus fails.\n\n**Weaknesses:**\nWhile I see that the paper seems like a nice theoretical contribution, I still think that the paper reads as something developed in a vacuum. More discussion of why these results are an important event, which is needed for the next step in a practically relevant application, would be a great addition. There is limited discussion on the impact of this particular result proved in the paper.\n\n- **Limited practical relevance:** The paper fails to connect theoretical insights to downhill ramification (practical algorithm design, impact on real-world applications, etc.).\n- **Narrow experimental validation:** Evaluation restricted to synthetic Gaussian Wigner experiments. The paper is missing evaluations on standard graph matching benchmarks or real-world datasets. Graph alignment has extensive applications in computational biology, social networks, and computer vision; however, there are no specific ablation studies on the impact of specific graph strucutre on the bounds, even empirically would be great to see. Helps talk about future directions.\n\nQUESTIONS:\n1. **Real-world relevance:** Is it a valid question to ask how the noise levels in your theoretical model relate to those encountered in practical applications mentioned in the introduction? What range of σ values is typical in practice? Are applications grouped based on where such a relaxation can be applied and where it cannot be?\n\n2. **Scope extension:** Can these techniques extend beyond the Gaussian Wigner model to more general graph alignment settings? What are the main technical challenges when you are thinking of, let us say, sub-Gaussian, Bernoulli, etc.? Is there practical relevance to these other extension models?\n\nLIMITATIONS:\nIt would be great if the authors added a dedicated limitations section discussing the points touched upon in the Weaknesses and Questions section.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 1, "ORIGINALITY": 4, "DATE": "2025-07-08"}