A complete dichotomy theorem on the sparse $t$-Uniform Hypergraphicality Problem
István Miklós, Miklós Ruszinkó, Bogdán Zavalnij
TL;DR
The paper studies the $t$-uniform hypergraphic degree sequence problem in a sparse, parameterized regime and establishes a complete dichotomy: 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 extends dense-regime hardness and dichotomy results to sparse instances, showing a sharp phase transition controlled by the minimum-to-maximum degree exponent ratio. The authors introduce gadget constructions and perturbation techniques (including LBDS and hinge-flips) to bridge realizability with a divisibility condition by $t$, yielding a unified framework across sparse and dense regimes. These results illuminate the role of sparsity in hypergraphical realizability and provide precise algorithmic boundaries for practical instance classes.
Abstract
We prove a complete dichotomy theorem for the parameterized sparse $t$-uniform hypergraphical 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 computational phase transition separating intractable and tractable regimes, fully characterizing the complexity of sparse $t$-uniform hypergraphicality across all degree exponents. 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 hypergraphical degree sequences.