Hypergraphic Degree Sequences are Hard

TL;DR

This paper proves that determining whether a vector can be realized as the degree sequence of a 3-hypergraph is an NP-complete problem, highlighting computational complexity in hypergraph theory.

Contribution

It establishes the NP-completeness of the hypergraphic degree sequence problem for 3-hypergraphs, a previously unresolved complexity question.

Findings

Deciding hypergraphic degree sequences is NP-complete.

The problem remains hard even for 3-hypergraphs.

Provides complexity classification for hypergraph degree sequences.

Abstract

We show that deciding if a given vector is the degree sequence of a 3-hypergraph is NP-complete.

Equations

$$w\sum S_{+}=w(c+\sum S_{+})=w\sum H=w\sum (H\cap S_{-})+w\sum (H\cap S_0)+w\sum (H\cap S_{+})$$

Full text

Hypergraphic Degree Sequences are Hard

Antoine Deza, Asaf Levin, Syed M. Meesum, and Shmuel Onn

A $k$-hypergraph on $[n]$ is a subset $H\subseteq\{0,1\}^{n}_{k}:=\{x\in\{0,1\}^{n}:\|x\|_{1}=k\}$. The degree sequence of $H$ is the vector $d=\sum H:=\sum\{x:x\in H\}$. We consider the following decision problem: given $k$ and $d\in\mathbb{Z}_{+}^{n}$, is $d$ the degree sequence of some hypergraph $H\subseteq\{0,1\}^{n}_{k}$? For $k=2$ (graphs) the celebrated work of Erdős and Gallai [3, 1960] implies that $d$ is a degree sequence of a graph if and only if $\sum d_{i}$ is even and, permuting $d$ so that $d_{1}\geq\dots\geq d_{n}$, the inequalities $\sum_{i=1}^{j}d_{i}-\sum_{i=l+1}^{n}d_{i}\leq j(l-1)$ hold for $1\leq j\leq l\leq n$, yielding a polynomial time algorithm. For $k=3$ the problem was raised over 30 years ago by Colbourn, Kocay, and Stinson [1, 1986, Problem 3.1] and was solved by Deza, Levin, Meesum, and Onn [2, 2018].

Here is the statement and its short proof.

Theorem: It is NP-complete to decide if $d\in\mathbb{Z}_{+}^{n}$ is the degree sequence of a $3$-hypergraph.

*Proof. *The problem is in NP since if $d$ is a degree sequences then a hypergraph $H\subseteq\{0,1\}^{n}_{3}$ of cardinality $|H|\leq{n\choose 3}=O(n^{3})$ can be exhibited and $d=\sum H$ verified in polynomial time.

Let ${\bf 1}$ be the all-ones vector. We consider the following three decision problems.

(1) Given $a\in\mathbb{Z}_{+}^{n}$, $b\in\mathbb{Z}_{+}$ with $3{\bf 1}a=nb$, is there $F\subseteq\{x\in\{0,1\}^{n}_{3}:ax=b\}$ with $\sum F={\bf 1}$ ?

(2) Given $w\in\mathbb{Z}^{n}$, $c\in\mathbb{Z}_{+}^{n}$ with $wc=0$, is there $G\subseteq\{x\in\{0,1\}^{n}_{3}:wx=0\}$ with $\sum G=c$ ?

(3) Given $d\in\mathbb{Z}_{+}^{n}$, is there $H\subseteq\{0,1\}^{n}_{3}$ with $\sum H=d$ ?

Problem (1) is the so-called $3$-partition problem which is known to be NP-complete [4].

First we reduce (1) to (2). Given $a,b$ with $3{\bf 1}a=nb$, let $w:=3a-b{\bf 1}$ and $c:={\bf 1}$. Then $wc=0$. Now, for any $x\in\{0,1\}^{n}_{3}$ we have $wx=3ax-b{\bf 1}x=3(ax-b)$ so $x$ satisfies $ax=b$ if and only if $wx=0$. So the answer to (1) is YES if and only if the answer to (2) is YES.

Second we reduce (2) to (3). Given $w,c$, with $wc=0$, define $d:=c+\sum S_{+}$, where $S_{\sigma}:=\{x\in\{0,1\}^{n}_{3}:{\rm sign}(wx)=\sigma\}$ for $\sigma=-,0,+$. Suppose there is a $G\subseteq S_{0}$ with $\sum G=c$. Then $H:=G\cup S_{+}$ satisfies $\sum H=d$. Suppose there is an $H\subseteq\{0,1\}^{n}_{3}$ with $\sum H=d$. Then

[TABLE]

which implies $H\cap S_{-}=\emptyset$ and $H\cap S_{+}=S_{+}$. Therefore $G:=H\cap S_{0}$ satisfies $\sum G=\sum H-\sum S_{+}=c$. So the answer to (2) is YES if and only if the answer to (3) is YES.