We study the $t$-uniform hypergraphicality problem under a compressed representation of the degree sequence. Instead of listing all vertex degrees explicitly, the input consists of pairs $$ (δ_1,n_1),\dots,(δ_k,n_k), $$ meaning that exactly $n_i$ vertices have degree $δ_i$. Thus the parameter $k$ denotes the number of distinct degrees. Although deciding $t$-hypergraphicality is NP-complete for every fixed $t>2$, we prove that the problem is fixed-parameter tractable parameterized by $(k,t)$. Our result shows that tractability extends substantially beyond previously known bounded-range regimes: even degree sequences with large overall degree spread can be handled efficiently when the number of distinct degrees is bounded. Our approach decomposes hyperedges according to their types with respect to the degree classes, yielding a bounded-dimension spectrum representation. Using balancing hinge-flips, we show that every feasible spectrum can be transformed into a realization of the prescribed degree sequence. This leads to an integer programming feasibility formulation with $$ \binom{t+k-1}{k-1} $$ variables. Applying Lenstra's theorem yields an FPT algorithm running in time $$ f(k,t)\cdot \mathrm{poly}(L), $$ where $L$ denotes the encoding length of the compressed input.

翻译：我们研究在度序列压缩表示下的$t$-均匀超图性问题。输入并非显式列出所有顶点的度数，而是由配对$$ (\delta_1,n_1),\dots,(\delta_k,n_k) $$ 构成，表示恰好有$n_i$个顶点具有度数$\delta_i$。因此参数$k$表示不同度数的数量。尽管对于每个固定的$t>2$，判定$t$-超图性问题是NP完全的，但我们证明该问题在参数化$(k,t)$下是固定参数可解的。我们的结果表明可解性显著扩展了先前已知的有界范围情形：即使整体度数跨度较大的度序列，在有限的不同度数数量下也能高效处理。我们的方法根据超边相对于度数类别的类型进行分解，从而得到有界维度的谱表示。通过平衡铰链翻转，我们证明每个可行的谱都可以转化为指定度序列的一个实现。这导致一个整数规划可行性公式，其变量数为$$ \binom{t+k-1}{k-1} $$。应用Lenstra定理得到运行时间为$$ f(k,t)\cdot \mathrm{poly}(L) $$的FPT算法，其中$L$表示压缩输入的编码长度。