Hardness of Forcing Unique Perfect Matchings in Bipartite Graphs of Maximum Degree 3
本文证明了在最大度为3的二分图中，通过强制集或反强制集使完美匹配唯一的问题是NP完全的。
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Stabilizing Private LASSO under Heterogeneous Covariates via Anisotropic Objective Perturbation
This work addresses the instability of objective perturbation in high-dimensional LASSO under differential privacy, which arises from heterogeneity in covariate scales and compromises both estimation accuracy and privacy guarantees. The authors propose an anisotropic objective perturbation method based on the Gram matrix, employing a “pre-distortion” strategy to counteract perturbation distortions induced by covariate structure and thereby restore isotropy in the estimation process. Notably, this approach directly incorporates structural information of the covariates into the perturbation mechanism, eliminating the need for privacy-budget-consuming data preprocessing. By integrating the algorithm within an approximate message passing (AMP) framework and leveraging state evolution analysis, the method achieves significantly improved convergence stability, statistical efficiency, and privacy performance while maintaining rigorous differential privacy guarantees.
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Meta Theorem for Hardness on FCP-Problem
This paper addresses the computational complexity of the Fewest Clues Problem (FCP) variant for NP-complete problems, specifically asking whether the FCP variant of every NP-complete problem is Σ₂^p-complete—a long-standing open question, previously resolved only for isolated cases. Method: We introduce the first general metatheorem that characterizes sufficient structural conditions for Σ₂^p-completeness of FCP variants, integrating polynomial-time reductions, second-order Boolean logic modeling, and constructive Σ₂^p-completeness techniques. Contribution/Results: We rigorously establish Σ₂^p-completeness for the FCP variants of canonical NP-complete problems—including SAT and Vertex Cover—and derive a reusable hardness certification framework. This resolves the open problem in full generality, providing both a foundational theoretical basis and practical tools for future analysis of FCP variants.
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Hardness of Forcing Unique Perfect Matchings in Bipartite Graphs of Maximum Degree 3
本文证明了在最大度为3的二分图中，通过强制集或反强制集使完美匹配唯一的问题是NP完全的。
0 citationsRead paper
Stabilizing Private LASSO under Heterogeneous Covariates via Anisotropic Objective Perturbation
This work addresses the instability of objective perturbation in high-dimensional LASSO under differential privacy, which arises from heterogeneity in covariate scales and compromises both estimation accuracy and privacy guarantees. The authors propose an anisotropic objective perturbation method based on the Gram matrix, employing a “pre-distortion” strategy to counteract perturbation distortions induced by covariate structure and thereby restore isotropy in the estimation process. Notably, this approach directly incorporates structural information of the covariates into the perturbation mechanism, eliminating the need for privacy-budget-consuming data preprocessing. By integrating the algorithm within an approximate message passing (AMP) framework and leveraging state evolution analysis, the method achieves significantly improved convergence stability, statistical efficiency, and privacy performance while maintaining rigorous differential privacy guarantees.
0 citationsRead paper
Meta Theorem for Hardness on FCP-Problem
This paper addresses the computational complexity of the Fewest Clues Problem (FCP) variant for NP-complete problems, specifically asking whether the FCP variant of every NP-complete problem is Σ₂^p-complete—a long-standing open question, previously resolved only for isolated cases. Method: We introduce the first general metatheorem that characterizes sufficient structural conditions for Σ₂^p-completeness of FCP variants, integrating polynomial-time reductions, second-order Boolean logic modeling, and constructive Σ₂^p-completeness techniques. Contribution/Results: We rigorously establish Σ₂^p-completeness for the FCP variants of canonical NP-complete problems—including SAT and Vertex Cover—and derive a reusable hardness certification framework. This resolves the open problem in full generality, providing both a foundational theoretical basis and practical tools for future analysis of FCP variants.
0 citationsRead paper