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Computing the combinatorial canonical form of a layered mixed matrix

Kazuo Murota, Mark Scharbrodt

1998
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Optimization Methods and Software
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The concept of mixed matrices was introduced by Murota-Iri [MI85] as a tool for describing discrete $\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}/\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}$ systems (see [Mu96] for exposition), and subsequently, the CCF of $\mathrm{L}\mathrm{M}$ -matrices was established by
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... m{I}\mathrm{r}\mathrm{i}$ -Nakamura [MIN87] and Murota [Mu87] . An efficient algorithm for computing the CCF was designed with matroid theoretical methods (submodular flow model). This algorithm, to be described in Section 2, operates in two phases; the first phase detects a maximal independent assignment in an auxiliary network, and the second phase finds the decomposition. In the present paper, we deal with practical computing of the CCF. Since engineering applications usually are large scale, it is important to identify typical characteristics in practical situations in order to significantly speed up the algorithm on top of its theoretical efficiency. In that line, based on the original algorithm, we will present practically faster versions which use simple but very effective procedures in order to improve solving the underlying independent assignment subproblem. Also, we discuss implementation strategies. We implemented the algorithm in the Mathematica language which is suitable especially for symbolic computation. The code is available via anonymous ftp (see Appendix A for details). 2. The original CCF-algorithm. The CCF of a layered mixed matrix can be computed by first identifying a maximum independent assignment in an associated bipartite graph, and then applying the ${\rm Min}-\mathrm{c}_{\mathrm{u}}\mathrm{t}-\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{P}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ to the resulting auxiliary network. This is based on the fact that the rank of a layered mixed matrix can be characterized by the minimum value of a certain submodular function that can be represented as the cut function of an independent assignment problem. In what follows, we describe the algorithm of [Mu93], while referring the reader to $[\mathrm{M}\mathrm{u}87][\mathrm{M}\mathrm{u}93]$ for theoretical backgrounds.

doi:10.1080/10556789808805720
fatcat:djgc6nfehzf6phvq2ynnfsi3k4