Branch-width of represented matroids in matrix multiplication time

Abstract

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 algorithm with running time $O_{k,\mathbb{F}}(n^2)+O(n^{\ddot{I}}‰)$ 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 $\ddot{I}‰<2.3714$ 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)], have cubic-time bottlenecks. Moreover, if the input matrix representation is given in standard form, our algorithm runs in $O_{k,\mathbb{F}}(n^2)$ time, since $O(n^{\ddot{I}}‰)$ 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 $O(mn\min (m,n{)}^{\ddot{I}‰-2})$. 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 in 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 $\hat{I}\text{c◯}(n^{\ddot{I}}‰)$-time, this implies that the overhead of $O(n^{\ddot{I}}‰)$ is unavoidable. We also show strengthenings of this observation to rule out some approximations under this assumption.

30 pages

Topics & keywords