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On Circuit Valuation of Matroids

✍ Scribed by Kazuo Murota; Akihisa Tamura

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 257 KB
Volume: 26
Category: Article
ISSN: 0196-8858
DOI: 10.1006/aama.2000.0716

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✦ Synopsis

The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of R ∪ -∞ -valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of R ∪ -∞ -valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids.

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