On complete classes of valuated matroids
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866912132984995840 |
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| author | Husić, Edin Loho, Georg Smith, Ben Végh, László A. |
| author_facet | Husić, Edin Loho, Georg Smith, Ben Végh, László A. |
| contents | We characterize a rich class of valuated matroids, called R-minor valuated matroids that includes the indicator functions of matroids, and is closed under operations such as taking minors, duality, and induction by network. We exhibit a family of valuated matroids that are not R-minor based on sparse paving matroids. Valuated matroids are inherently related to gross substitute valuations in mathematical economics. By the same token we refute the Matroid Based Valuation Conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015) asserting that every gross substitute valuation arises from weighted matroid rank functions by repeated applications of merge and endowment operations. Our result also has implications in the context of Lorentzian polynomials: it reveals the limitations of known construction operations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_06961 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On complete classes of valuated matroids Husić, Edin Loho, Georg Smith, Ben Végh, László A. Combinatorics Discrete Mathematics Computer Science and Game Theory Optimization and Control We characterize a rich class of valuated matroids, called R-minor valuated matroids that includes the indicator functions of matroids, and is closed under operations such as taking minors, duality, and induction by network. We exhibit a family of valuated matroids that are not R-minor based on sparse paving matroids. Valuated matroids are inherently related to gross substitute valuations in mathematical economics. By the same token we refute the Matroid Based Valuation Conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015) asserting that every gross substitute valuation arises from weighted matroid rank functions by repeated applications of merge and endowment operations. Our result also has implications in the context of Lorentzian polynomials: it reveals the limitations of known construction operations. |
| title | On complete classes of valuated matroids |
| topic | Combinatorics Discrete Mathematics Computer Science and Game Theory Optimization and Control |
| url | https://arxiv.org/abs/2107.06961 |