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In this paper, we consider the problem of computing the entire sequence ofthe maximum degree of minors of a block-structured symbolic matrix (a genericpartitioned polynomial matrix) $A = (A_{\alpha\beta} x_{\alpha \beta}t^{d_{\alpha \beta}})$, where $A_{\alpha\beta}$ is a $2 \times 2$ matrix over afield $\mathbf{F}$, $x_{\alpha \beta}$ 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 beviewed as an algebraic generalization of the maximum weight bipartite matchingproblem.  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 maximumdegree of minors of a $(2 \times 2)$-type generic partitioned polynomial matrixof size $2\mu \times 2\nu$. We also present a minimax theorem, which can beused as a good characterization (NP $\cap$ co-NP characterization) for thecomputation of the maximum degree of minors of order $k$. Our resultsgeneralize the classical primal-dual algorithm (the Hungarian method) andminimax formula (Egerv\'ary's theorem) for the maximum weight bipartitematching problem.

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