A dichotomy theorem on the complexity of 3-uniform hypergraphic degree sequence graphicality
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2024
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| author | Logsdon, Sara Maheshwari, Arya Miklós, István Zhang, Angelina |
| author_facet | Logsdon, Sara Maheshwari, Arya Miklós, István Zhang, Angelina |
| contents | We present a dichotomy theorem on the parameterized complexity of the 3-uniform hypergraphicality problem. Given $0<c_1\le c_2 < 1$, the parameterized 3-uniform Hypergraphic Degree Sequence problem, $3uni-HDS_{c_1,c_2}$, considers degree sequences $D$ of length $n$ such that all degrees are between $c_1 {n-1 \choose 2}$ and $c_2 {n-1\choose 2}$ and it asks if there is a 3-uniform hypergraph with degree sequence $D$. We prove that for any $0<c_2< 1$, there exists a unique, polynomial-time computable $c_1^*$ with the following properties. For any $ c_1\in (c_1^*,c_2]$, $3uni-HDS_{c_1,c_2}$ can be solved in linear time. In fact, for any $c_1\in (c_1^*,c_2]$ there exists an easy-to-compute $n_0$ such that any degree sequence $D$ of length $n\ge n_0$ and all degrees between $c_1 {n-1\choose 2}$ and $c_2 {n-1\choose 2}$ has a 3-uniform hypergraph realization if and only if the sum of the degrees can be divided by $3$. Further, $n_0$ grows polynomially with the inverse of $c_1-c_1^*$. On the other hand, we prove that for all $c_1<c_1^*$, $3uni-HDS_{c_1,c_2}$ is NP-complete. Finally, we briefly consider an extension of the hypergraphicality problem to arbitrary $t$-uniformity. We show that the interval where degree sequences (satisfying divisibility conditions) always have $t$-uniform hypergraph realizations must become increasingly narrow, with interval width tending to $0$ as $t \rightarrow \infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19049 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A dichotomy theorem on the complexity of 3-uniform hypergraphic degree sequence graphicality Logsdon, Sara Maheshwari, Arya Miklós, István Zhang, Angelina Combinatorics MSC 2020 05C65, 05C07, 05C85, 68Q17, 68Q27 We present a dichotomy theorem on the parameterized complexity of the 3-uniform hypergraphicality problem. Given $0<c_1\le c_2 < 1$, the parameterized 3-uniform Hypergraphic Degree Sequence problem, $3uni-HDS_{c_1,c_2}$, considers degree sequences $D$ of length $n$ such that all degrees are between $c_1 {n-1 \choose 2}$ and $c_2 {n-1\choose 2}$ and it asks if there is a 3-uniform hypergraph with degree sequence $D$. We prove that for any $0<c_2< 1$, there exists a unique, polynomial-time computable $c_1^*$ with the following properties. For any $ c_1\in (c_1^*,c_2]$, $3uni-HDS_{c_1,c_2}$ can be solved in linear time. In fact, for any $c_1\in (c_1^*,c_2]$ there exists an easy-to-compute $n_0$ such that any degree sequence $D$ of length $n\ge n_0$ and all degrees between $c_1 {n-1\choose 2}$ and $c_2 {n-1\choose 2}$ has a 3-uniform hypergraph realization if and only if the sum of the degrees can be divided by $3$. Further, $n_0$ grows polynomially with the inverse of $c_1-c_1^*$. On the other hand, we prove that for all $c_1<c_1^*$, $3uni-HDS_{c_1,c_2}$ is NP-complete. Finally, we briefly consider an extension of the hypergraphicality problem to arbitrary $t$-uniformity. We show that the interval where degree sequences (satisfying divisibility conditions) always have $t$-uniform hypergraph realizations must become increasingly narrow, with interval width tending to $0$ as $t \rightarrow \infty$. |
| title | A dichotomy theorem on the complexity of 3-uniform hypergraphic degree sequence graphicality |
| topic | Combinatorics MSC 2020 05C65, 05C07, 05C85, 68Q17, 68Q27 |
| url | https://arxiv.org/abs/2411.19049 |