A complete dichotomy theorem on the sparse $t$-Uniform Hypergraphicality Problem

Abstract

We prove a complete dichotomy theorem for the parameterized sparse $t$-uniform hypergraphic degree sequence problem, $\mathrm{sparse}\text{-}t\text{-}\mathrm{uni}\text{-}\mathrm{HDS}_{\alpha',\alpha}$. For any fixed $t \ge 3$, given parameters $0 \le \alpha' \le \alpha < t-1$, the input consists of degree sequences $D$ of length $n$ with degrees between $n^{\alpha'}$ and $6n^{\alpha}$. We show that the problem is NP-complete whenever $\alpha' \le \frac{t(\alpha - 1) + 1}{t - 1}$, and solvable in linear time when $\alpha' > \frac{t(\alpha - 1) + 1}{t - 1}$. 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 $\Omega(n^{\frac{t-1}{t}})$) 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.