Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

Maximizing Products of Linear Forms, and The Permanent of Positive Semidefinite Matrices

We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite (HPSD) matrices. By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to the permanent of HPSD matrices, as well as computationally efficient certificates for this approximation. We also propose an analog of van der Waerden's conjecture for HPSD matrices, where the polynomial optimization problem i…

关于矩阵积和式的两个猜想之间的逻辑蕴含关系

A logical implication between two conjectures on matrix permanents

本研究证明了Bapat与Sunder提出的两个半正定矩阵积和式猜想间存在逻辑蕴含，既推导了满足首个猜想的矩阵类也满足第二个，还通过构造第二个猜想的新反例得到首个猜想的新反例，建立了两条件的非平凡关联。

关于图拉普拉斯矩阵的霍尔特猜想

Structural Classes for Chollet's Permanent Conjecture

本文研究图拉普拉斯矩阵的永久不等式，通过顶点合并框架证明了霍尔特不等式在特定条件下的成立，扩展了不等式应用范围，揭示了永久性计算的新可计算领域。

伴随矩阵的秩-零度定理的永久性类比

Permanental Analog of the Rank-Nullity Theorem for Symmetric Matrices

本文提出永久性秩-零度定理的类比，针对对称非负矩阵、正半定矩阵和平衡符号图邻接矩阵，证明其永久性秩与零度之和等于n，并在多项式时间内计算永久性零度。

永久性秩与随机矩阵在有限域上的行列式秩

Permanental rank versus determinantal rank of random matrices over finite fields

研究随机矩阵在有限域上永久性秩与行列式秩的区别，发现永久性秩概率与零列概率相似，而行列式秩概率则不同。

A polynomial time algorithm to approximate the mixed volume within a simply exponential factor

Let ${\bf K} = (K_1, ..., K_n)$ be an $n$-tuple of convex compact subsets in the Euclidean space $\R^n$, and let $V(\cdot)$ be the Euclidean volume in $\R^n$. The Minkowski polynomial $V_{\bf K}$ is defined as $V_{\bf K}(λ_1, ... ,λ_n) = V(λ_1 K_1 +, ..., + λ_n K_n)$ and the mixed volume $V(K_1, ..., K_n)$ as $$ V(K_1, ..., K_n) = \frac{\partial^n}{\partial λ_1...\partial λ_n} V_{\bf K}(λ_1 K_1 +, ..., + λ_n K_n). $$ Our main result is a poly-time algorithm which approximates $V(K_1, ..., K_n)$ with multiplicative error $e^n$ and with better rates if the affine dimensions of most of the sets $…