一种计算行列式阶的成本定标算法
A cost-scaling algorithm for computing the degree of determinants
摘要：
本文讨论了\[ A = \sum_{k=1}^m A_k x_k t^{c_k}, \]的Dieudonne行列式$\mathop{\rm Det} A$的次数$\mathop{\rm deg Det} A$的计算，其中$A_k$是域$\mathbb{K}$上的$n \times n$矩阵，$x_k$是非交换变量，$t$是与$x_k$可交换的变量，$c_k$是整数，并考虑了$t$的次数.该问题推广了非交换Edmonds问题和基本组合优化问题，包括加权线性拟阵求交问题。结果表明，$\mathop{\rm deg Det} A$是通过一个离散的凸优化的欧几里德建筑。我们扩展了这个框架，将成本缩放技术，并表明，$\mathop{\rm deg Det} A$可以计算在时间多项式的$n,m,\log_2 C$，其中$C:= \max_k |c_k|$。我们给出了一个多面体的解释$\mathop{\rm deg Det}$，它说，$\mathop{\rm deg Det} A$是由线性优化的整数多面体关于目标向量$c = (c_k)$。在此基础上，我们证明了我们的算法成为一个强多项式的。我们将这一结果应用到代数组合优化问题所产生的符号矩阵具有$2 \times 2$ -子矩阵结构。
英文摘要：
In this paper, we address computation of the degree $\mathop{\rm deg Det} A$ of Dieudonné determinant $\mathop{\rm Det} A$ of \[ A = \sum_{k=1}^m A_k x_k t^{c_k}, \] where $A_k$ are $n \times n$ matrices over a field $\mathbb{K}$, $x_k$ are noncommutative variables, $t$ is a variable commuting with $x_k$, $c_k$ are integers, and the degree is considered for $t$. This problem generalizes noncommutative Edmonds' problem and fundamental combinatorial optimization problems including the weighted linear matroid intersection problem. It was shown that $\mathop{\rm deg Det} A$ is obtained by a discrete convex optimization on a Euclidean building. We extend this framework by incorporating a cost scaling technique, and show that $\mathop{\rm deg Det} A$ can be computed in time polynomial of $n,m,\log_2 C$, where $C:= \max_k |c_k|$. We give a polyhedral interpretation of $\mathop{\rm deg Det}$, which says that $\mathop{\rm deg Det} A$ is given by linear optimization over an integral polytope with respect to objective vector $c = (c_k)$. Based on it, we show that our algorithm becomes a strongly polynomial one. We apply this result to an algebraic combinatorial optimization problem arising from a symbolic matrix having $2 \times 2$-submatrix structure.
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