Fixed-Parameter Tractability of $t$-Uniform Hypergraphicality
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 $$ (\delta_1,n_1),\dots,(\delta_k,n_k), $$ meaning that exactly $n_i$ vertices have degree $\delta_i$. 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.