We address the computation of the degrees of minors of a noncommutative symbolic matrix of form \[ A[c] := \sum_{k=1}^m A_k t^{c_k} x_k, \] where $A_k$ are matrices over a field $\mathbb{K}$, $x_i$ are noncommutative variables, $c_k$ are integer weights, and $t$ is a commuting variable specifying the degree. This problem extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can formulate various combinatorial optimization problems. Extending the study by Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and polyhedral characterization for the maximum degrees of minors of $A[c]$ of all sizes, and develop a strongly polynomial-time algorithm for computing them. This algorithm is viewed as a unified algebraization of the classical Hungarian method for bipartite matching and the weight-splitting algorithm for linear matroid intersection. As applications, we provide polynomial-time algorithms for weighted fractional linear matroid matching and linear optimization over rank-2 Brascamp-Lieb polytopes.

翻译：我们研究形如 \[ A[c] := \sum_{k=1}^m A_k t^{c_k} x_k \] 的非交换符号矩阵的子式次数计算问题，其中 $A_k$ 为域 $\mathbb{K}$ 上的矩阵，$x_i$ 为非交换变量，$c_k$ 为整数权重，$t$ 为指定次数的交换变量。该问题推广了非交换埃德蒙兹问题（Ivanyos 等，2017），并可形式化描述多种组合优化问题。本文延续 Hirai（2018）及 Hirai、Ikeda（2022）的研究，提出了关于 $A[c]$ 所有大小子式最大次数的对偶定理与多面体刻画，并开发了用于计算这些最大次数的强多项式时间算法。该算法可视为经典二部匹配匈牙利算法与线性拟阵交权重分裂算法的统一代数化形式。作为应用，我们给出了加权分数线性拟阵匹配与秩-2 Brascamp-Lieb 多胞体上线性优化的多项式时间算法。