Branch-width of represented matroids in matrix multiplication time

TL;DR

This paper introduces a faster algorithm for computing the branch-width of matroids represented by matrices over finite fields, improving over previous cubic-time algorithms and approaching matrix multiplication time.

Contribution

It presents an algorithm with nearly quadratic time complexity for branch-width computation, leveraging matrix multiplication and standard form representations, with applications to related graph parameters.

Findings

New algorithm runs in O(n^2) time for standard form matrices.

Achieves near matrix multiplication time complexity for branch-width computation.

Provides faster algorithms for rank-width and path-width in related structures.

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 $(O_{k,\mathbb F}(n^2)+O(n^\omega))$-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 $\omega < 2.3714$ 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 Hlin\v{e}n\'y and Oum [SIAM J. Comput. (2008)] and Jeong, Kim, and Oum [SIAM J. Discrete Math. (2021)] run in at least $\Omega(n^3)$ 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 $O(n^\omega)$-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 $O(mn \min(m,n)^{\omega-2})$. 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 $\Omega(n^\omega)$-time, this implies that the overhead of $O(n^{\omega})$ is unavoidable. We also show strengthenings of this observation to rule out some approximations under this assumption.