Integer points in the degree-sequence polytope

Abstract

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