A primitive with a sharper error dependence on $n$ and $d$ for the downstream cut approximation is developed and combined with a new edge-sensitive terminal cut oracle with additive error yields the final worst-case polynomial-time private cut-release error.
Abstract
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.
A spectral sparsifier of a weighted graph is a reweighted subgraph whose Laplacian quadratic form approximates that of the original graph. Let $G$ be a positively weighted $n$-vertex, $m$-edge multigraph, let $0<\varepsilon\le1/2$. Assuming $m,\varepsilon^{-1}\le n^{O(1)}$ and the ratio of maximum to minimum weight is polynomially bounded, we deterministically construct a $(1\pm\varepsilon)$-spectral sparsifier with \[ O\!\left(n\varepsilon^{-2}\log^{24+o(1)}n\right) \] edges in \[ m^{1+o(1)}+O\!\left(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n\right) \] time. The construction has two main ingredients. First, we sparsify an approximately regular expander by partitioning its edges into few matchings and viewing their normalized Laplacians as an isotropic family of positive semidefinite matrices. Rather than sample from this family and apply matrix Chernoff, we select matchings deterministically using a pessimistic estimator. We evaluate the resulting conditional-expectation scores in two ways to produce two algorithms: using dense matrix multiplication and sparsely using polynomial approximations to the inverse square root and matrix exponential. Deterministic expander decomposition, along with replacing vertices by fixed expander graphs to achieve approximate regularity, extends these algorithms to general graphs. Second, a recursive blocking scheme applies the dense algorithm to smaller subgraphs and the sparse algorithm to their union, balancing their costs. Reusing the resulting algorithm as the dense algorithm gives $\alpha_{r+1}=3-1/(\alpha_r-1)$, starting from $\alpha_0=\omega$. After $O(\log n)$ levels, the exponent is $2+O(1/\log n)$, yielding $m^{1+o(1)}+\widetilde O_{\varepsilon}(n^2)$ time.
For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollob\'as--Erd\H{o}s--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed positive linear degree, and, within every hereditary graph class, a uniform sublinear bound is equivalent to a sublinear bound on graphs of every fixed positive linear vertex connectivity. We prove the sharp general estimate \[ h(G)\le \left\lfloor\frac{|V(G)|}{2\alpha(G)+\delta(G)-|V(G)|}\right\rfloor \] whenever the denominator is positive, with equality for balanced complete multipartite graphs. Consequently, every $3$-colorable graph of order $n$ with $\kappa(G)\ge\rho n$ and $\rho>1/3$ has a hitting set of size at most $\lfloor(\rho-1/3)^{-1}\rfloor$; direct use of a $3$-coloring improves this to $6$ when $\kappa(G)>4n/9$ and to the sharp bound $3$ when $\kappa(G)>n/2$. For dense regular graphs with independence ratio greater than $1/4$, we obtain a logarithmic bound, while constructions with linear degree and linear independence number show that $h(G)=\Omega(\sqrt n)$ can still occur. We also prove a logarithmic bound for near-regular $3$-colorable graphs and exhibit a critical family at connectivity $n/3$ that explains the limitations of the degree-surplus and degree-ratio methods.
This paper provides the first truly linear-time approximation scheme for the Densest Subgraph Problem, and uses assignments arising from a flow-based formulation together with a structural carving lemma to progressively carve "sparse" parts of the graph while nearly preserving the densest subgraph.
For a simple graph $G$ of order $n$, let $S_2(G)=\lambda_1(G)+\lambda_2(G)$ denote its spectral sum. We determine, for every $n\geq5$, the exact maximum of $S_2(G)$ and all equality cases. The unique maximizer, up to isomorphism, is the complement of the disjoint union of a suitably balanced complete bipartite graph and isolated vertices, with the sizes of its three parts determined by $n$ modulo $7$. Denoting this graph by $K_n^\star$, we further show that $ S_2(K_n^\star)\leq\frac{8n}{7}-2,$ with equality exactly when $7\mid n$. This proves a conjecture of Kumar, Liu, Monterde, Pragada and Tait, which strengthens the Aouchiche--Hansen 2010 conjecture by extending it from connected graphs to all graphs and by asserting uniqueness of the extremal graph. The result also subsumes the 2008 conjecture of Ebrahimi B., Mohar, Nikiforov, and Ahmady. The proof combines Ky Fan's variational principle with a spectral inequality for weighted Ferrers quotients to reduce the problem to an explicit family whose complements have incidence rank one. Exact integer optimization and a separate equality analysis then yield the maximum and uniqueness.
A randomized data structure for undirected weighted graphs that are partially dynamic, i.e., that undergo either only edge insertions or only edge deletions is given, which follows from a simple stability principle for partially dynamic graphs.
Gramoz Goranci, Rasmus Kyng, Maximilian Probst Gutenberg et al.· 0 citations
Motivated by recent work of Kocurek, Oveis Gharan, and Tjowasi, which gives an efficient sampling algorithm for the hard-core model on random regular bipartite graphs by decomposing into fixed-size slices, we study the worst-case tractability of approximate counting and sampling of fixed-size slices for bipartite independent set problems. Let $G=(L\sqcup R,E)$ be a bipartite graph with $|L|=|R|=n$ and maximum degree $\Delta$. The fixed-slice problem asks to sample uniformly from independent sets satisfying $|I\cap L|=\alpha_L n$ and $|I\cap R|=\alpha_R n$. We show that if the overall density $\alpha$ lies in the interval $(\frac{1}{\Delta}, \tfrac{1}{2})$, and the densities on the two sides are more balanced than the typical phase densities of a random $\Delta$-regular bipartite graph, then there is no FPRAS or efficient sampling scheme unless $\mathbf{NP}=\mathbf{RP}$. We then study a related fugacity model in which the densities are not fixed, but the independent set is required to be balanced between the two sides of the bipartition. For $\lambda>0$, the balanced hard-core model is the ordinary hard-core model with fugacity $\lambda$, conditioned on the event $|I\cap L|=|I\cap R|$. We prove that this model has the same computational threshold as the hard-core model on general bounded-degree graphs. That is, for every fixed $\Delta\ge 3$, if $\lambda<\lambda_c(\Delta)$, then the balanced partition function admits an FPTAS and the balanced hard-core distribution admits an efficient sampling scheme. Conversely, if $\lambda>\lambda_c(\Delta)$, then no FPRAS or efficient sampler exists on this graph class unless $\mathbf{NP}=\mathbf{RP}$.
Ijay Narang, Will Perkins, Yuzhou Wang et al.· 1 citation
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