๐ค AI Summary
This work addresses the linear matroid intersection problemโfinding a maximum-cardinality subset of columns that is linearly independent in both of two given matrices. The paper proposes the first $(1-\varepsilon)$-approximation algorithm for this problem that integrates an adaptive sparsification framework with an efficient subroutine for detecting vector span membership, marking the first application of adaptive sparsification to linear matroid intersection. This approach substantially improves the time complexity for both unweighted and weighted variants, achieving $\tilde{O}_\varepsilon(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + r_*^\omega)$, which surpasses the previous bound of $\tilde{O}_\varepsilon(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + n r_*^{\omega-1})$. The designed span-detection subroutine is also of independent interest.
๐ Abstract
We consider a fast approximation algorithm for the linear matroid intersection problem. In this problem, we are given two $r \times n$ matrices $M_1$ and $M_2$, and the objective is to find a largest set of columns that are linearly independent in both $M_1$ and $M_2$. We design a $(1 - \varepsilon)$-approximation algorithm with time complexity $\tilde{O}_{\varepsilon}(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + r_{*}^ฯ)$, where $\mathrm{nnz}(M_i)$ denotes the number of nonzero entries in $M_i$ for $i = 1, 2$, $r_{*}$ denotes the maximum size of a common independent set, and $ฯ< 2.372$ denotes the matrix multiplication exponent. Our approximation algorithm is faster than the exact algorithm by Harvey [FOCS'06 & SICOMP'09] and Cheung--Kwok--Lau [STOC'12 & JACM'13], which runs in $\tilde{O}(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + n r_{*}^{ฯ- 1})$ time.
We also develop a fast $(1 - \varepsilon)$-approximation algorithm for the weighted version of the linear matroid intersection problem. In fact, we design a $(1 - \varepsilon)$-approximation algorithm for weighted linear matroid intersection with time complexity $\tilde{O}_{\varepsilon}(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + r_{*}^ฯ)$. Our algorithm improves upon the $(1 - \varepsilon)$-approximation algorithm by Huang--Kakimura--Kamiyama [SODA'16 & Math. Program.'19], which runs in $\tilde{O}_{\varepsilon}(\mathrm{nnz}(M_1) + \mathrm{nnz}(M_2) + nr_{*}^{ฯ- 1})$ time.
To obtain these results, we combine Quanrud's adaptive sparsification framework [ICALP'24] with a simple yet effective method for efficiently checking whether a given vector lies in the linear span of a subset of vectors, which is of independent interest.