Hypergraph Degree Sequence Complexity

• The paper establishes NP-completeness for k-uniform hypergraph degree sequences (k≥3) and delineates phase boundaries between intractable and tractable regimes.
• It shows that constant, almost-regular, and dense irregular sequences can be realized in polynomial time using combinatorial designs and Havel–Hakimi–style algorithms.
• Randomized algorithms and MCMC methods are integrated with algebraic approaches via toric ideals, linking combinatorial optimization with statistical model complexity.

A hypergraphical degree sequence specifies, for a fixed integer $k \geq 2$, a list $D = (d_1, \ldots, d_n)$ of nonnegative integers, where $d_i$ is the desired number of $k$-element hyperedges incident to vertex $i$ in a simple $k$-uniform hypergraph $H$ on $n$ vertices. The core computational question is whether such an $H$ exists (the decision problem) and, if so, how to efficiently construct or count such realizations. The complexity landscape for these problems is sharply stratified by the degree distribution and uniformity parameter $k$, exhibiting a rich structure of computational phase transitions, algorithmic barriers, and tractable subregions. Recent work has delineated these boundaries with increasing precision.

1. Problem Formulation and Core Complexity Barriers

The $k$-Uniform Hypergraph Degree Sequence (k-HDS) problem asks: Given $n$, $k$, and $D\in\mathbb N^n$, does there exist a simple $k$-uniform hypergraph $H$ with $\deg_H(i) = d_i$ for all $i$? For $k = 2$ (graphs), the classical Erdős–Gallai and Havel–Hakimi criteria provide a polynomial-time decision and construction algorithm. For $k\geq3$, the situation fundamentally changes: Deza, Levin, Meesum, and Onn established that deciding $k$-HDS is NP-complete for every fixed $k\geq3$ (Deza et al., 2019, Deza et al., 2017, Bhattacharya et al., 2021). NP-membership follows as a solution can be guessed and checked in $O(n^k)$ time, but the hardness arises via strong reductions (notably from 3-Partition): the presence of a degree sequence encoding an instance of a classic NP-complete problem ensures worst-case intractability.

This NP-completeness extends to various generalizations: partite $k$-uniform hypergraphs (Hubai et al., 2023), line graph–based reconstruction (Bhattacharya et al., 2021), and even for optimizing (e.g., quadratic) objective functions over simple $k$-hypergraphs (Deza et al., 2017). Thus, the default regime for arbitrary degree sequences is computationally infeasible in the classical sense.

2. Tractable Subregions: Dense, Regular, and Balanced Sequences

Despite the overarching hardness, large islands of polynomial-time tractability persist under severe restrictions:

• Constant or Nearly-Constant Sequences: If $D$ is constant ($d_1 = \dots = d_n = d$), existence reduces to verifying $k \mid n d$, realizability is guaranteed, and explicit construction follows by Baranyai’s theorem using combinatorial design methods in time $O(n^{3k+1})$ for fixed $k$ (Bhattacharya et al., 2021).
• Almost-Regular Sequences: If degrees differ by at most 1 (e.g., $d_i\in\{k, k+1\}$) and $\sum d_i \equiv 0\pmod{k}$, polynomial-time Havel–Hakimi–style procedures can efficiently construct a realizing hypergraph (Li et al., 2023).
• Dense Irregular Windows: For $k=3$, if all degrees $$d_i\in[\frac{2}{63}n^2 + O(n),\,\frac{5}{63}n^2 - O(n)]$$ with $n\geq45$ and $\sum d_i \equiv 0 \pmod{3}$, one can construct a realizing hypergraph in time $O(n^3)$; this extends the known tractable regime to highly irregular, dense degree windows (Li et al., 2023).
• Partite Hypergraphs (Restricted Classes): In 3-partite, 3-uniform hypergraphs, if the degree sequence in one vertex class is "third almost-regular" ($k$ or $k-1$), realization and construction are polynomial-time (Hubai et al., 2023).

These P-regions are globally atypical; almost all sequences fall into the hard regime except when severe balancing, regularity, or density window constraints are imposed.

3. Parameterized and Phase Transition Results

Recent advances have constructed sharp dichotomy theorems specifying the critical thresholds—or "phase boundaries"—separating polynomial and NP-complete regimes:

Three-Uniform: Degree Window Dichotomy

For 3-uniform hypergraphs, specifying lower and upper normalized degree bounds $c_1\binom{n-1}{2} \leq d_i \leq c_2\binom{n-1}{2}$, there exists an explicit function $c_1^*(c_2)$ such that

• For $c_1 > c_1^*(c_2)$ and $n > n_0(c_1, c_2)$, every degree sequence satisfying the window and modulus condition ($\sum d_i \equiv 0 \pmod{3}$) is always graphic, and the problem is solvable in linear time.
• For $c_1 < c_1^*(c_2)$, even permitting $d_i$ slightly below $c_1^*(c_2)$, $3$-uni-HDS is NP-complete (Logsdon et al., 28 Nov 2024).

The threshold $c_1^*$ is determined by explicit combinatorial optimization over extremal constructions, and $n_0$ is polynomial in $1/(c_1-c_1^*)$. The tractable regime is underpinned by constructive rounds and Havel–Hakimi–like hinge-flip operations.

Sparse $t$-Uniform: Exponent Dichotomy

For arbitrary $t\geq3$, one considers the sparse-$t$-uni-HDS$_{\alpha',\alpha}$ problem: $n^{\alpha'}\leq d_i\leq 6 n^\alpha$. There is a sharp phase transition at $\alpha^* = \frac{t(\alpha-1)+1}{t-1}$:

• If $\alpha' > \alpha^*$, every such degree sequence is graphic as soon as $\sum d_i$ is divisible by $t$, and realization is linear-time (Miklós et al., 17 Dec 2025).
• If $\alpha' \leq \alpha^*$, the problem is NP-complete, even for moderate density and sparsity ranges.

This result unifies and interpolates between dense and sparse dichotomies and generalizes the Erdős–Gallai threshold for graphs to all $t \geq 3$.

Table 1: Tractability Thresholds for Hypergraphical Degree Sequences

Uniformity $k$, regime	Tractable (P)	Hard (NP-complete)
$k=2$ (graphs)	all sequences	--
$k\geq3$, general	--	all sequences
$k=3$, window $c_1^*<c_1\leq d_i/\binom{n-1}{2}\leq c_2<1$	window	below window
$t\geq3$, degree exponents $\alpha',\alpha$	$\alpha' > \alpha^*$	$\alpha' \leq \alpha^*$

Here, witness algorithms exist in the "P" regions, and constructive realization is always possible.

4. Randomized and Markov Chain Algorithms

Algorithmic approaches outside the classical window combine randomized construction, bipartite reductions, and Markov Chain Monte Carlo (MCMC):

• Randomized Ball-into-Box Allocations: For $k$-uniform sequences with $(k+1) d_{k+2} \leq o = \sum d_i$, there is a randomized polynomial-time procedure outputting a simple realizing hypergraph with strictly positive (whp approaching 1 as $n\to\infty$ under mild balance) probability (Ascolese et al., 7 Feb 2024). The approach greedily allocates stubs to boxes before sampling edges, carefully controlling loops and parallel edges.
• Bipartite Rejection Sampling: Using the correspondence between $k$-uniform hypergraphs and bipartite graphs with degree constraints, one can sample nearly uniformly at random from the set of simple hypergraphs with a given degree sequence via a rejection sampling algorithm, provided sparsity/regularity conditions yield sufficiently high acceptance probability. The cost scales as $O(1/q)$, where $q$ is the probability a random bipartite realization is "hypergraph-simple." Uniformity is retained for suitable degree ranges and decreases exponentially as degrees grow (Dyer et al., 2020).
• Markov Chains and Parallel Tempering: For partite and general uniform hypergraphs, MCMC (switch, hinge-flip, simulated annealing) are proposed to sample and mix over the realization space. Polynomial-time mixing is established only in certain "stable" or "third almost-regular" classes. Heuristic methods such as parallel tempering perform well in practice but lack worst-case guarantees outside the tractable regime (Hubai et al., 2023, Dyer et al., 2020).

5. Extension to Algebraic and Statistical Complexity

The hypergraphical degree sequence problem and associated realization/Markov chain moves admit an algebraic formulation: the toric ideal $I_H$ associated to $H$ encodes all polynomial relations among degree-preserving moves. The degree complexity of Markov bases—minimal generating binomials—connects combinatorial gadgetry (splitting sets, walks) to the size and structure of statistical models:

• For $k$-partite uniform hypergraphs, Markov width is bounded ($\leq 2$) and Markov bases are small (polynomial in $|E|$).
• In general, if $H$ contains large primitive walks with no splitting sets, Markov degree complexity grows linearly or worse, reflecting wild behavior in sampling and model fit (Gross et al., 2012).

This complexity propagates into algebraic statistics (e.g., contingency tables, hidden subset models), where boundaries between feasible and infeasible Markov moves are determined by the same combinatorial phase transitions.

6. Open Questions and Frontier Directions

Several cutoffs and transition curves have been precisely established, but the landscape remains punctuated by unresolved phenomena:

• The exact minimal constant $c$ for which any $D$ with $d_i\in[c n^{k-1}, (1-c) n^{k-1}]$ is always $k$-graphic is unknown except for $k=2$ (Li et al., 2023).
• The necessity and sufficiency of sparsity/regularity conditions (e.g., $k^2 d_{\max} = o(M)$) for polynomial expected success of randomized or rejection sampling is not fully characterized, and the boundary for uniform simple hypergraph sampling is open (Dyer et al., 2020).
• The existence and sharpness of "gap" regions between easy and hard regimes, especially for highly irregular and moderately dense degree sequences, is an ongoing area of research (Miklós et al., 17 Dec 2025, Li et al., 2023).
• The complexity for degree sequences with bounded support or mixtures of small and large entries, and the effect of additional constraints (fixed $k$ as variable, linearity, bounded pair-multiplicity), remain open.
• Markov chain mixing times, the practical uniformity of combinatorial MCMC, and algebraic degree complexity for further classes (e.g., $t$-way tables), are major challenges bridging combinatorics, optimization, and statistics (Gross et al., 2012, Hubai et al., 2023).

7. Synthesis and Summary

The complexity landscape for hypergraphical degree sequences is dominated by a "hard core" (NP-completeness for $k\geq3$) interrupted by sharply delimited P-regions corresponding to regularity, density, or balance conditions. Tight dichotomy theorems (Logsdon et al., 28 Nov 2024, Miklós et al., 17 Dec 2025) have now mapped these boundaries across both dense and sparse regimes and for increasing uniformities $t$. Randomized and MCMC-based construction algorithms bridge a further gap, efficiently realizable whenever the degree distribution skirts certain thresholds, but fundamentally bounded by combinatorial and algebraic constraints. The study of Markov bases and toric ideals imports algebraic and statistical perspectives, unifying combinatorics, complexity, and applications in modeling and sampling. The location and characterization of transition boundaries, especially in intermediate and irregular regimes, represent central open questions in contemporary hypergraph theory and related fields.