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| Formato: | Preprint |
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2025
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| Acceso en línea: | https://arxiv.org/abs/2512.15356 |
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| _version_ | 1866917173154283520 |
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| author | Miklós, István Ruszinkó, Miklós Zavalnij, Bogdán |
| author_facet | Miklós, István Ruszinkó, Miklós Zavalnij, Bogdán |
| contents | We prove a complete dichotomy theorem for the parameterized sparse $t$-uniform hypergraphic degree sequence problem, $\mathrm{sparse}\text{-}t\text{-}\mathrm{uni}\text{-}\mathrm{HDS}_{α',α}$. For any fixed $t \ge 3$, given parameters $0 \le α' \le α< t-1$, the input consists of degree sequences $D$ of length $n$ with degrees between $n^{α'}$ and $6n^α$. We show that the problem is NP-complete whenever $α' \le \frac{t(α- 1) + 1}{t - 1}$, and solvable in linear time when $α' > \frac{t(α- 1) + 1}{t - 1}$. This establishes a sharp boundary between polynomial-time solvable and NP-complete instances, thereby characterizing the computational complexity across all degree exponent regimes. The result extends the earlier NP-completeness of dense hypergraphicality to a unified framework covering both sparse and dense regimes, revealing that even extremely sparse instances (with maximum degree $o(n)$ but $Ω(n^{\frac{t-1}{t}})$) remain NP-complete. On the other hand, the $t$-uniform hypergraphicality solvable in linear time when the maximum degree is $o(n^{\frac{t-1}{t}})$. This dichotomy provides a comprehensive classification of the complexity landscape for hypergraphic degree sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15356 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A complete dichotomy theorem on the sparse $t$-Uniform Hypergraphicality Problem Miklós, István Ruszinkó, Miklós Zavalnij, Bogdán Combinatorics We prove a complete dichotomy theorem for the parameterized sparse $t$-uniform hypergraphic degree sequence problem, $\mathrm{sparse}\text{-}t\text{-}\mathrm{uni}\text{-}\mathrm{HDS}_{α',α}$. For any fixed $t \ge 3$, given parameters $0 \le α' \le α< t-1$, the input consists of degree sequences $D$ of length $n$ with degrees between $n^{α'}$ and $6n^α$. We show that the problem is NP-complete whenever $α' \le \frac{t(α- 1) + 1}{t - 1}$, and solvable in linear time when $α' > \frac{t(α- 1) + 1}{t - 1}$. This establishes a sharp boundary between polynomial-time solvable and NP-complete instances, thereby characterizing the computational complexity across all degree exponent regimes. The result extends the earlier NP-completeness of dense hypergraphicality to a unified framework covering both sparse and dense regimes, revealing that even extremely sparse instances (with maximum degree $o(n)$ but $Ω(n^{\frac{t-1}{t}})$) remain NP-complete. On the other hand, the $t$-uniform hypergraphicality solvable in linear time when the maximum degree is $o(n^{\frac{t-1}{t}})$. This dichotomy provides a comprehensive classification of the complexity landscape for hypergraphic degree sequences. |
| title | A complete dichotomy theorem on the sparse $t$-Uniform Hypergraphicality Problem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.15356 |