Unimodular polytopes and column number bounds on polytopal totally unimodular matrices via Seymour's decomposition theorem
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917402149650432 |
|---|---|
| author | Nill, Benjamin |
| author_facet | Nill, Benjamin |
| contents | We prove a sharp upper bound on the number of distinct columns of a totally unimodular matrix with column sums $1$ improving upon Heller's classical bound. The proof uses Seymour's decomposition theorem. Such matrices are closely related to unimodular polytopes: lattice polytopes where the vertices of every full-dimensional subsimplex form an affine lattice basis. This is an interesting subclass of 0/1-polytopes and contains for instance edge polytopes of bipartite graphs. Our main result on totally unimodular matrices implies a sharp upper bound on the number of vertices of unimodular polytopes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13431 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unimodular polytopes and column number bounds on polytopal totally unimodular matrices via Seymour's decomposition theorem Nill, Benjamin Combinatorics Optimization and Control 52B12, 52B20, 90C10 We prove a sharp upper bound on the number of distinct columns of a totally unimodular matrix with column sums $1$ improving upon Heller's classical bound. The proof uses Seymour's decomposition theorem. Such matrices are closely related to unimodular polytopes: lattice polytopes where the vertices of every full-dimensional subsimplex form an affine lattice basis. This is an interesting subclass of 0/1-polytopes and contains for instance edge polytopes of bipartite graphs. Our main result on totally unimodular matrices implies a sharp upper bound on the number of vertices of unimodular polytopes. |
| title | Unimodular polytopes and column number bounds on polytopal totally unimodular matrices via Seymour's decomposition theorem |
| topic | Combinatorics Optimization and Control 52B12, 52B20, 90C10 |
| url | https://arxiv.org/abs/2405.13431 |