发表机构
Graduate School of Mathematics, Nagoya University（名古屋大学数学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Edmonds问题中线性符号矩阵秩计算的多面体方面，建立整数多面体层级并证明整数间隙下界，对秩2反对称矩阵改进间隙至2/3。
AI 中文摘要
本文研究了交换与非交换Edmonds问题中，用于计算线性符号矩阵$A = \sum_{k=1}^m A_k x_k$秩的多面体方面。我们将这些问题分别视为在整数多面体${\cal P}(A)$和${\cal Q}(A)$上的线性优化，这些多面体由$A$的子行列式指数向量及其膨胀$A^{\{d\}} = \sum_{k=1}^m A_k \otimes X_k$（$d=1,2,\ldots$）的凸包获得。通过扩展先前关于nc-秩的已知结果，我们建立了整数多面体的层级${\cal P}(A) \subseteq {\cal P}^{\leq 2}(A) \subseteq {\cal P}^{\leq 3}(A) \subseteq \cdots = {\cal Q}(A)$，并证明了${\cal Q}(A)$相对于${\cal P}(A)$的整数间隙至少为$1/2$。进一步，我们证明了如果每个$A_k$是秩为2的反对称矩阵，则上述层级在第二层终止，且整数间隙改进为$2/3$。
英文摘要
In this paper, we study polyhedral aspects on commutative and noncommutative Edmonds' problems for computing the rank of linear symbolic matrix $A = \sum_{k=1}^m A_k x_k$. We regard them as linear optimization over integral polytopes ${\cal P}(A)$ and ${\cal Q}(A)$, respectively, which are obtained by the convex hulls of exponent vectors of subdeterminants of~$A$ and its blow-ups $A^{\{d\}} = \sum_{k=1}^m A_k \otimes X_k$ $(d=1,2,\ldots)$. By extending previously known results on nc-rank, we establish a hierarchy of integral polytopes ${\cal P}(A) \subseteq {\cal P}^{\leq 2}(A) \subseteq {\cal P}^{\leq 3}(A) \subseteq \cdots = {\cal Q}(A)$ and show that the integrality gap of ${\cal Q}(A)$ relative to ${\cal P}(A)$ is at least $1/2$. Further, we show that if each $A_k$ is rank-2 skew-symmetric, then the above hierarchy terminates at the second level and the integrality gap is improved to $2/3$.
