An integer vector $b \in \mathbb{Z}^d$ is a degree sequence if there exists a hypergraph with vertices $\{1,\dots,d\}$ such that each $b_i$ is the number of hyperedges containing $i$. The degree-sequence polytope $\mathscr{Z}^d$ is the convex hull of all degree sequences. We show that all but a $2^{-\Omega(d)}$ fraction of integer vectors in the degree sequence polytope are degree sequences. Furthermore, the corresponding hypergraph of these points can be computed in time $2^{O(d)}$ via linear programming techniques. This is substantially faster than the $2^{O(d^2)}$ running time of the current-best algorithm for the degree-sequence problem. We also show that for $d\geq 98$, the degree-sequence polytope $\mathscr{Z}^d$ contains integer points that are not degree sequences. Furthermore, we prove that the linear optimization problem over $\mathscr{Z}^d$ is $\mathrm{NP}$-hard. This complements a recent result of Deza et al. (2018) who provide an algorithm that is polynomial in $d$ and the number of hyperedges.

翻译：一个整数向量 $b \in \mathbb{Z}^d$ 若存在一个顶点集为 $\{1,\dots,d\}$ 的超图，使得每个 $b_i$ 是包含顶点 $i$ 的超边数量，则称其为度序列。度序列多面体 $\mathscr{Z}^d$ 是所有度序列的凸包。我们证明，在度序列多面体中的整数向量中，除比例为 $2^{-\Omega(d)}$ 的部分外，其余均为度序列。此外，这些点对应的超图可通过线性规划技巧在 $2^{O(d)}$ 时间内计算得到，这比当前度序列问题最优算法的 $2^{O(d^2)}$ 运行时间快得多。我们还证明，对于 $d\geq 98$，度序列多面体 $\mathscr{Z}^d$ 包含非度序列的整数点。进一步，我们证明在 $\mathscr{Z}^d$ 上的线性优化问题是 $\mathrm{NP}$-难的。这一结果补充了 Deza 等人 (2018) 的近期工作，他们提出了一个关于 $d$ 和超边数量均为多项式时间的算法。