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

In this paper, we consider the problem of computing the degree of the determinant 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 $\alpha, \beta = 1,2,\dots, n$, and $t$ is an additional indeterminate. This problem can be viewed as an algebraic generalization of the maximum weight perfect bipartite matching problem. The main result of this paper is a combinatorial $O(n^4)$-time algorithm for the deg-det computation of a $(2 \times 2)$-type generic partitioned polynomial matrix of size $2n \times 2n$. We also present a min-max theorem between the degree of the determinant and a potential defined on vector spaces. Our results generalize the classical primal-dual algorithm (Hungarian method) and min-max formula (Egerv\'ary's theorem) for maximum weight perfect bipartite matching.