A (convex) polytope P is said to be 2-level if every hyperplane H that is facet-defining for P has a parallel hyperplane H' that contains all the vertices of P which are not contained in H.Two level polytopes appear in different areas of mathematics, in particular in contexts related to discrete geometry and optimization. We study the problem of enumerating all combinatorial types of 2-level polytopes of a fixed dimension d. We describe the first algorithm to achieve this. We ran it to produce the complete database for d <= 8. Our results show that the number of combinatorial types of 2-level d-polytopes is surprisingly small for low dimensions d.We provide an upper bound for the number of combinatorially inequivalent 2-level d-polytopes. We phrase this counting problem in terms of counting some objects called 2-level configurations, that capture the class of "maximal" rank d 0/1-matrices, including (maximal) slack matrices of 2-level cones and 2-level polytopes. We provide a proof that the number of d-dimensional 2-level configurations coming from cones and polytopes, up to linear equivalence, is at most 2^{O(d^2 log d)}.Finally, we prove that the extension complexity of every stable set polytope of a bipartite graph with n nodes is O(n^2 log n) and that there exists an infinite class of bipartite graphs such that, for every n-node graph in this class, its stable set polytope has extension complexity equal to Omega(n log n). / Doctorat en Sciences / info:eu-repo/semantics/nonPublished
| Identifer | oai:union.ndltd.org:ulb.ac.be/oai:dipot.ulb.ac.be:2013/276475 |
| Date | 07 September 2018 |
| Creators | Macchia, Marco |
| Contributors | Fiorini, Samuel, Cardinal, Jean, D'Adderio, Michele, Padrol, Arnau, Sanyal, Raman |
| Publisher | Universite Libre de Bruxelles, Université libre de Bruxelles, Faculté des Sciences – Mathématiques, Bruxelles |
| Source Sets | Université libre de Bruxelles |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/doctoralThesis, info:ulb-repo/semantics/doctoralThesis, info:ulb-repo/semantics/openurl/vlink-dissertation |
| Format | No full-text files |
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