A combinatorial algorithm for computing the entire sequence of the maximum degree of minors of a generic partitioned polynomial matrix with $2 \times 2$ submatrices

In this paper, we consider the problem of computing the entire sequence of the maximum degree of minors of a block-structured symbolic matrix (a generic partitioned polynomial matrix) $A = (A_{\alpha\beta} x_{\alpha \beta} t^{d_{\alpha \beta}})$, where $A_{\alpha\beta}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$, $x_{\alpha \beta}$ is an indeterminate, and $d_{\alpha \beta}$ is an integer for $\alpha = 1,2,\dots, \mu$ and $\beta = 1,2,\dots,\nu$, and $t$ is an additional indeterminate. This problem can be viewed as an algebraic generalization of the maximum weight bipartite matching problem. The main result of this paper is a combinatorial $O(\mu \nu \min\{\mu, \nu\}^2)$-time algorithm for computing the entire sequence of the maximum degree of minors of a $(2 \times 2)$-type generic partitioned polynomial matrix of size $2\mu \times 2\nu$. We also present a minimax theorem, which can be used as a good characterization (NP $\cap$ co-NP characterization) for the computation of the maximum degree of minors of order $k$. Our results generalize the classical primal-dual algorithm (the Hungarian method) and minimax formula (Egerv\'ary's theorem) for the maximum weight bipartite matching problem.