A Combinatorial Algorithm for Computing the Rank of a Generic Partitioned Matrix with 2 × 2 Submatrices

Abstract

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) [Formula Presented], where [Formula Presented] is a [Formula Presented] matrix over a field [Formula Presented] and [Formula Presented] is an indeterminate for [Formula Presented] and [Formula Presented]. 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 combinatorial [Formula Presented]-time algorithm for computing the symbolic rank of a [Formula Presented]-type generic partitioned matrix of size [Formula Presented]. Our algorithm is based on the Wong sequence algorithm by Ivanyos, Qiao, and Subrahamanyam for the nc-rank of a general symbolic matrix, but is simpler. Our proposed algorithm 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 [Formula Presented].