Private Approximation of Graph Spectra and Cuts via Spectral Amplifiers
We study the problem of releasing a synthetic graph that approximates the sizes of all cuts of an input graph under edge-level differential privacy. If one insists on purely additive error, the optimal worst-case error is $\widetilde\Theta(n^{3/2})$. If one allows a small multiplicative slack, an information-theoretic exponential-time mechanism achieves nearly linear additive error, but the best known polynomial-time algorithms have substantially larger error. We give a polynomial-time $(\varepsilon,\delta)$-differentially private algorithm which, for every $n$-vertex unweighted graph $G$, outputs a non-negative weighted synthetic graph $\widetilde G$ such that, with high probability, every cut $S\subseteq V(G)$ satisfies \[ |w_G(S)-w_{\widetilde G}(S)| \le \gamma w_G(S)+\widetilde O_{\varepsilon,\delta,\gamma}(n^{13/12+o(1)}). \] This improves the previous polynomial-time worst-case bound $\widetilde O(n^{5/4+o(1)})$ of Aamand et al. (ICML 2025) for mixed multiplicative/additive private cut approximation. The main technical ingredient is a new set of private spectral primitives for bounded-degree graphs, one of them gives spectral error $\widetilde O_{\delta}((nd)^{1/4}/\sqrt\varepsilon)$ in estimating the graph Laplacian for graphs of maximum degree $d$, being the first to beat the standard $\min\{2d,\widetilde O_{\delta}(\sqrt{n}/\varepsilon)\}$ baseline in the high-degree regime. We further develop a primitive with a sharper error dependence on $n$ and $d$ for the downstream cut approximation. Combined with a new edge-sensitive terminal cut oracle with additive error $\widetilde O(n+(n^2M)^{1/3})$ on graphs with $M$ edges, this yields the final worst-case $\widetilde O(n^{13/12+o(1)})$ private cut-release error.