sdp subgraph recovery

This paper addresses exact community recovery in the asymmetric binary Stochastic Block Model (SBM). While existing semidefinite programming (SDP) methods achieve the information-theoretic limit under symmetry, their failure mechanism in the asymmetric setting remains unclear, and no principled SDP design exists for this case. First, the authors uncover the geometric root of symmetric SDP failure: it provably fails to recover communities exactly in certain information-theoretically solvable regimes, exposing a fundamental limitation of standard SDP formulations. Second, they propose a novel adaptive SDP framework that explicitly incorporates asymmetry via prior constraints on community sizes and a weighted objective function reflecting heterogeneous edge probabilities. Theoretically, the new SDP achieves exact recovery over a strictly broader parameter regime and attains the tight information-theoretic threshold. This work extends the applicability boundary of SDP in statistical inference and establishes a new paradigm for learning from structurally imbalanced graphs.

Understanding the interplay between association schemes and Lovász–Schrijver lift-and-project relaxations—particularly SDP-based ones—in combinatorial optimization, especially for stable set polytopes and hypergraph matching. Method: We introduce the notion of *deeply vertex-transitive graphs*, construct *hypermatching pseudo-schemes* (noncommutative association structures), and develop a systematic contraction method to derive commutative sub-schemes. Our approach integrates algebraic combinatorics, spectral graph theory, and SDP analysis. Contribution/Results: We derive Delsarte–Hoffman-type tight bounds on clique and stability numbers; precisely characterize the lift-and-project rank of hypergraph matching relaxations; and establish a general contraction framework from noncommutative pseudo-schemes to commutative sub-schemes. This yields a unified algebraic–convex optimization paradigm for symmetric combinatorial optimization problems.

This work addresses the exact recovery of planted cliques in semi-random graphs. We consider the Feige–Kilian-type semi-random model under adversarial edge perturbations and propose a novel semidefinite programming (SDP) algorithm. Our method introduces a tailored rectangular constraint rounding scheme and develops refined structural analysis techniques for the SDP solution. We prove that the algorithm achieves exact recovery of the planted clique with high probability (w.h.p.), attaining the same detection threshold as the optimal result in purely random graphs. Moreover, it resolves a key special case of the DkSReg model studied in KL20. To our knowledge, this is the first work to establish tight theoretical guarantees for exact recovery in a non-uniform semi-random setting, significantly extending both the applicability and robustness of existing SDP-based approaches.

This paper addresses the Sparse Integer Least Squares (SILS) problem—NP-hard least squares optimization under {0, ±1} constraints and sparsity. To tackle it, we propose the first ℓ₁-regularized semidefinite programming (SDP) relaxation and design a complementary randomized rounding algorithm. Theoretically, we derive a sufficient condition for exact recovery via SDP under fixed sparsity, leveraging sub-Gaussian analysis and second-moment characterization of the covariance matrix to establish exact recovery even under weakly correlated features. Empirically, we validate the method’s effectiveness and practicality across diverse applications, including privacy-preserving identification, multi-user detection, feature extraction, and integer sparse signal recovery. Our approach bridges rigorous theoretical guarantees—providing provable exact recovery under realistic statistical assumptions—with computational tractability and real-world deployability.

This work addresses the optimization landscape analysis of the Burer–Monteiro (BM) low-rank factorization method for MaxCut-type semidefinite programs (SDPs). We identify the condition number of the associated Laplacian matrix as a key spectral criterion: when this condition number falls below a tight, explicitly characterized threshold, every second-order critical point of the nonconvex BM formulation is guaranteed to be a global optimum, and no spurious local minima exist. This constitutes the first tight sufficient condition ensuring global convergence of the BM method for MaxCut-type SDPs, markedly improving the theoretical solvability boundary for canonical problems such as ℤ₂-synchronization. Technically, we integrate tools from Riemannian optimization, spectral graph theory, and second-order critical point analysis to establish a precise quantitative relationship between the Laplacian condition number and the benignness of the optimization landscape. Our results provide a more rigorous theoretical foundation for solving nonconvex low-rank SDPs.

本文解决了最大割问题中SDP精确性识别的复杂性问题，通过构造具有多项式界整数权重的平方和对偶证书证明了加权图和简单无权图的强NP难性。

This study addresses the unresolved tightness bounds and error exponents of semidefinite programming (SDP) for community recovery when the number of communities grows logarithmically. Focusing on the balanced stochastic block model, it integrates SDP relaxation, spectral analysis, and concentration inequalities to reveal that a local correction mechanism intrinsically governs both tightness and error exponents. The work derives sharp asymptotic tightness bounds alongside matching high-probability error exponents. Furthermore, it demonstrates that exact recovery is achievable through a single SDP solve even with a suboptimal planted matrix, establishing theoretical guarantees above the information-theoretic threshold and confirming robustness against perturbations from rare vertices.

This work proposes a novel self-supervised graph neural network architecture to reduce the substantial computational cost of repeatedly solving Max-Cut semidefinite programming (SDP) relaxations within branch-and-bound algorithms. Designed as a lightweight SDP proxy solver embedded in an exact optimization framework, the model requires no ground-truth SDP solutions for supervision and directly predicts primal- and dual-feasible SDP solutions. These predictions are combined with Goemans–Williamson randomized rounding to produce high-quality cut solutions. To the best of our knowledge, this is the first graph neural network approach that achieves fully self-supervised training while preserving feasibility, significantly accelerating the solution of structured convex relaxations. Experiments demonstrate that the method reduces the boundary computation cost in exact Max-Cut solving by up to 10.6× compared to the commercial solver MOSEK.

This study addresses the problem of recovering a planted $k$-vertex subgraph with elevated triangle density in an Erdős–Rényi random graph—a task known to be computationally hard in the worst case. To tackle this challenge, the work introduces spectral and semidefinite programming algorithms based on a novel local signed triangle count matrix, marking the first incorporation of signed triangle counts into a spectral recovery framework. Theoretical analysis establishes that the information-theoretic threshold for exact recovery scales logarithmically in $n$, whereas the proposed algorithms succeed only when $k$ is at least on the order of $\sqrt{n}$, thereby revealing a substantial statistical–computational gap. The paper also provides rigorous recovery guarantees for the algorithms and validates their performance within the low-degree polynomial framework.

This study addresses the absence of a unified continuous modeling and compilation framework for discrete NP combinatorial optimization problems. To this end, it proposes a unified paradigm that reformulates discrete NP problems as continuous standard quadratic programs (StQPs). Building upon a simplex framework, the work integrates graph reductions, regularized Motzkin–Straus formulations, and QUBO mapping techniques to construct a coefficient-bounded finite-domain factor model compiler, requiring at most four simplex coordinates per binary factor. The primary contribution is the exact, standardized StQP compilation of Karp’s 21 NP-complete problems, accompanied by explicit formulas, dimensionality analyses, and separation bounds. This formulation rigorously guarantees that all valid assignments correspond to global or local minima, thereby establishing a complete continuous solution pathway for combinatorial optimization.