The $r$-coloring and maximum stable set problem in hypergraphs with bounded matching number and edge size

TL;DR

This paper investigates the computational complexity of coloring and stable set problems in hypergraphs with bounded matching number and edge size, providing dichotomies, algorithms, and hardness results.

Contribution

It offers a comprehensive complexity classification for various hypergraph coloring and stable set problems with bounded matching number and edge size, including new polynomial-time algorithms and NP-hardness proofs.

Findings

Complete dichotomies for hypergraph coloring and stable set problems.

Polynomial-time algorithm for 2-coloring certain 3-uniform hypergraphs.

NP-hardness of 3-coloring linear 3-uniform hypergraphs with bounded matching number.

Abstract

Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting $\nu(G)$ denote the maximum size of a matching in $H$, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed $r, k, s \in \mathbb{N}$: $r$-Coloring in hypergraphs $G$ with edge size at most $k$ and $\nu(G) \leq s$; $r$-Precoloring Extension in $k$-uniform hypergraphs $G$ with $\nu(G) \leq s$; $r$-Precoloring Extension in hypergraphs $G$ with edge size at most $k$ and $\nu(G) \leq s$; Maximum Stable Set in $k$-uniform hypergraphs $G$ with $\nu(G) \leq s$; Maximum Weight Stable Set in $k$-uniform hypergraphs with $\nu(G) \leq s$; as well as partial results for $r$-Coloring in $k$-uniform hypergraphs $\nu(G) \leq s$. We then turn our attention to $2$-Coloring in 3-uniform hypergraphs with forbidden induced subhypergraphs, and give a polynomial-time algorithm when restricting the input to hypergraphs excluding a fixed one-edge hypergraph. Finally, we consider linear 3-uniform hypergraphs (in which every two edges share at most one vertex), and show that excluding an induced matching in $G$ implies that $\nu(G)$ is bounded by a constant; and that $3$-coloring linear $3$-uniform hypergraphs $G$ with $\nu(G) \leq 532$ is NP-hard.