For an $n$-element matroid $M$ given by an $n \times n$ matrix representation over a finite field $\mathbb F$ and an integer $k$, we present an $(O_{k,\mathbb F}(n^2)+O(n^ω))$-time algorithm that either finds a branch-decomposition of $M$ of width at most $k$, or confirms that the branch-width of $M$ is more than $k$, where $ω< 2.3714$ is the matrix multiplication exponent, and the $O_{k,\mathbb F}(\cdot)$-notation hides factors that depend on $k$ and $\mathbb F$ in a computable manner. All previous algorithms including Hliněný and Oum [SIAM J. Comput. (2008)] and Jeong, Kim, and Oum [SIAM J. Discrete Math. (2021)] run in at least $Ω(n^3)$ time. Moreover, if the input matrix representation is given by a standard form, our algorithm runs in $O_{k,\mathbb F}(n^2)$-time, since $O(n^ω)$-time is only needed for finding a standard form of the input matrix. When $M$ is given by an $m \times n$ matrix, the overhead for finding a standard form is $O(mn \min(m,n)^{ω-2})$. As corollaries, we obtain faster algorithms for rank-width of directed graphs and path-width of matroids represented over a fixed finite field. Furthermore, we also present an approximation algorithm for finding branch-width that works on infinite fields provided that the input matrix is of a standard form and contains a bounded number of distinct values of entries. To suggest that our algorithm is optimal, we observe that for every field $\mathbb F$, deciding whether the branch-width of a matroid represented over $\mathbb F$ is $0$ is as hard as deciding whether a square matrix over $\mathbb F$ is singular. Under the assumption that singularity testing requires $Ω(n^ω)$-time, this implies that the overhead of $O(n^ω)$ is unavoidable. We also show strengthenings of this observation to rule out some approximations under this assumption.

翻译：对于一个由有限域 $\mathbb F$ 上的 $n \times n$ 矩阵表示给定的 $n$ 元拟阵 $M$ 以及整数 $k$，我们提出了一种时间复杂度为 $(O_{k,\mathbb F}(n^2)+O(n^ω))$ 的算法。该算法要么找到 $M$ 的一个宽度不超过 $k$ 的分支分解，要么确认 $M$ 的分支宽度大于 $k$；其中 $ω< 2.3714$ 是矩阵乘法指数，而 $O_{k,\mathbb F}(\cdot)$ 记号隐藏了可计算地依赖于 $k$ 和 $\mathbb F$ 的因子。此前所有算法（包括 Hliněný 与 Oum [SIAM J. Comput. (2008)] 以及 Jeong, Kim 与 Oum [SIAM J. Discrete Math. (2021)]）的运行时间至少为 $Ω(n^3)$。此外，若输入矩阵表示以标准形式给出，我们的算法只需 $O_{k,\mathbb F}(n^2)$ 时间，因为 $O(n^ω)$ 时间仅用于寻找输入矩阵的标准形式。当 $M$ 由 $m \times n$ 矩阵给定时，寻找标准形式的额外开销为 $O(mn \min(m,n)^{ω-2})$。作为推论，我们获得了有向图的秩宽度以及固定有限域上表示拟阵的路径宽度的更快算法。此外，我们还提出了一种近似算法用于寻找分支宽度，该算法适用于无限域，前提是输入矩阵为标准形式且包含有界数量的不同条目值。为表明我们算法的最优性，我们注意到：对于每个域 $\mathbb F$，判定 $\mathbb F$ 上表示拟阵的分支宽度是否为 $0$ 的难度与判定 $\mathbb F$ 上矩阵是否奇异的难度相同。在假设奇异性检验需要 $Ω(n^ω)$ 时间的条件下，这意味着 $O(n^ω)$ 的额外开销是不可避免的。我们还给出了该观测结果的加强形式，以在此假设下排除某些近似算法。