A combinatorial algorithm for computing the rank of a generic partitioned matrix with $2 \times 2$ submatrices

Hiroshi Hirai, Yuni Iwamasa

Published 2020-04-22Version 1

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) $A = (A_{\alpha \beta} x_{\alpha \beta})$, where $A_{\alpha \beta}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$ and $x_{\alpha \beta}$ is an indeterminate for $\alpha = 1,2,\dots, \mu$ and $\beta = 1,2, \dots, \nu$. This problem can be viewed as an algebraic generalization of the bipartite matching problem, and was considered by Iwata and Murota (1995). One of recent interests on this problem lies in the connection with non-commutative Edmonds' problem by Ivanyos, Qiao, and Subrahamanyam (2018), and Garg, Gurvits, Oliveiva, and Wigderson (2019), where a result by Iwata and Murota implicitly says that the rank and the non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial $O((\mu \nu)^2 \min \{ \mu, \nu \})$-time algorithm for computing the symbolic rank of a $(2 \times 2)$-type generic partitioned matrix of size $2\mu \times 2\nu$. Our algorithm is inspired by the Wong sequence algorithm by Ivanyos, Qiao, and Subrahamanyam for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of $A$ for an arbitrary field $\mathbf{F}$.