Lower bounds and integrality gaps in simplicial decomposition
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929299634782208 |
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| author | Ellison, Matthew |
| author_facet | Ellison, Matthew |
| contents | Let $\mathcal{K}$ be a finite pure simplicial $d$-complex, with oriented facets $\{F_i\}$, which is boundaryless in the sense that $\sum\partial F_i=0$. We call such a $\mathcal{K}$ an \textit{admissible $d$-complex}. Given an admissible $d$-complex, one can ask for the smallest collection $\{T_i\}$ of oriented $(d+1)$-simplices on the vertices of $\mathcal{K}$ which decomposes $\mathcal{K}$ in the sense that $\sum \partial T_i = \mathcal{K}$. Let the minimum size of such a collection be $V_\mathbb{Z}(\mathcal{K})$, and let $V_\mathbb{Q}(\mathcal{K})$ be the relaxed analog where fractional $(d+1)$-simplices may be used. We explain how these quantities may be computed via integer and linear programming, and show how lower bounds may be obtained by exploiting LP-duality. We then prove that $V_\mathbb{Q}$ and $V_\mathbb{Z}$ are both additive under disjoint union and connected sum along a $d$-simplex. The remainder of the paper explores integrality gaps between $V_\mathbb{Z}$ and $V_\mathbb{Q}$ in dimension 1, where we share what we believe is the simplest admissible complex with an integrality gap; and in dimension 2, where we collect some results on integrality gaps for triangulations of the 2-sphere for a companion paper with Zili Wang and Peter Doyle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower bounds and integrality gaps in simplicial decomposition Ellison, Matthew Algebraic Topology Combinatorics 55U10, 52-08, 52C05, 52C07, 90C05, 90C10, 68W25 F.2.2 Let $\mathcal{K}$ be a finite pure simplicial $d$-complex, with oriented facets $\{F_i\}$, which is boundaryless in the sense that $\sum\partial F_i=0$. We call such a $\mathcal{K}$ an \textit{admissible $d$-complex}. Given an admissible $d$-complex, one can ask for the smallest collection $\{T_i\}$ of oriented $(d+1)$-simplices on the vertices of $\mathcal{K}$ which decomposes $\mathcal{K}$ in the sense that $\sum \partial T_i = \mathcal{K}$. Let the minimum size of such a collection be $V_\mathbb{Z}(\mathcal{K})$, and let $V_\mathbb{Q}(\mathcal{K})$ be the relaxed analog where fractional $(d+1)$-simplices may be used. We explain how these quantities may be computed via integer and linear programming, and show how lower bounds may be obtained by exploiting LP-duality. We then prove that $V_\mathbb{Q}$ and $V_\mathbb{Z}$ are both additive under disjoint union and connected sum along a $d$-simplex. The remainder of the paper explores integrality gaps between $V_\mathbb{Z}$ and $V_\mathbb{Q}$ in dimension 1, where we share what we believe is the simplest admissible complex with an integrality gap; and in dimension 2, where we collect some results on integrality gaps for triangulations of the 2-sphere for a companion paper with Zili Wang and Peter Doyle. |
| title | Lower bounds and integrality gaps in simplicial decomposition |
| topic | Algebraic Topology Combinatorics 55U10, 52-08, 52C05, 52C07, 90C05, 90C10, 68W25 F.2.2 |
| url | https://arxiv.org/abs/2404.01279 |