Computing Puiseux-series solutions to determinantal equations via combinatorial relaxation
SIAM Journal on Computing
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
Computing the Degree of Determinants via Combinatorial Relaxation
SIAM Journal on Computing
Primal-dual combinatorial relaxation algorithms for the maximum degree of subdeterminants
SIAM Journal on Scientific Computing
Valuated Matroid Intersection I: Optimality Criteria
SIAM Journal on Discrete Mathematics
Valuated Matroid Intersection II: Algorithms
SIAM Journal on Discrete Mathematics
On the Degree of Mixed Polynomial Matrices
SIAM Journal on Matrix Analysis and Applications
Maximum Matchings via Gaussian Elimination
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Maximum weight bipartite matching in matrix multiplication time
Theoretical Computer Science
Algebraic Algorithms for Matching and Matroid Problems
SIAM Journal on Computing
Matrices and Matroids for Systems Analysis
Matrices and Matroids for Systems Analysis
On the Kronecker Canonical Form of Mixed Matrix Pencils
SIAM Journal on Matrix Analysis and Applications
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Mixed polynomial matrices are polynomial matrices with two kinds of nonzero coefficients: fixed constants that account for conservation laws and independent parameters that represent physical characteristics. The computation of their maximum degrees of minors is known to be reducible to valuated independent assignment problems, which can be solved by polynomial numbers of additions, subtractions, and multiplications of rational functions. However, these arithmetic operations on rational functions are much more expensive than those on constants. In this paper, we present a new algorithm of combinatorial relaxation type. The algorithm finds a combinatorial estimate of the maximum degree by solving a weighted bipartite matching problem, and checks if the estimate is equal to the true value by solving independent matching problems. The algorithm mainly relies on fast combinatorial algorithms and performs numerical computation only when necessary. In addition, it requires no arithmetic operations on rational functions. As a byproduct, this method yields a new algorithm for solving a linear valuated independent assignment problem.