Graph Alignment via Birkhoff Relaxation
Sushil Mahavir Varma, Irène Waldspurger, Laurent Massoulié
TL;DR
This work analyzes the Birkhoff relaxation for graph alignment under a Gaussian Wigner model, proving a phase-transition in how close the relaxation’s solution $X^*$ is to the true permutation $\Pi^*$ as the noise parameter $\sigma$ grows. In particular, $\|X^*-\Pi^*\|_F^2 = o(n)$ when $\sigma=o(n^{-1})$, enabling 1-$o(1)$ vertex recovery via rounding, while $\|X^*-\Pi^*\|_F^2 = \Omega(n)$ when $\sigma=\Omega(n^{-0.5})$, indicating strong separation and signaling limits of direct recovery. The authors develop a dual-certificate-based proof, leverage GOE concentration, and provide simulations showing superior performance of the Birkhoff relaxation relative to GRAMPA and Simplex relaxations, including practical rounding with the Hungarian method. The results advance theoretical understanding of tight convex relaxations for QAP-like problems and suggest robustness of Birkhoff relaxation with post-processing beyond the proven regime. Overall, the paper offers state-of-the-art guarantees for permutation recovery in noisy graph alignment and highlights directions for improving thresholds and extending to broader noise models.
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 $1/\sqrt{1+σ^2}$. Denote the optimal solutions of the QAP and Birkhoff relaxation by $Π^\star$ and $X^\star$ respectively. We show that $\|X^\star-Π^\star\|_F^2 = o(n)$ when $σ= o(n^{-1.25})$ and $\|X^\star-Π^\star\|_F^2 = Ω(n)$ when $σ= Ω(n^{-0.5})$. Thus, the optimal solution $X^\star$ transitions from a small perturbation of $Π^\star$ for small $σ$ to being well separated from $Π^\star$ as $σ$ 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 $σ= o(n^{-1.25})$. This condition on $σ$ to ensure the success of the Birkhoff relaxation is state-of-the-art.