Improved Inapproximability of Rainbow Coloring

TL;DR

This paper establishes new NP-hardness results for coloring hypergraphs under rainbow coloring constraints, extending previous hardness results to more general and tighter parameters.

Contribution

It proves NP-hardness of 2-coloring in rainbow hypergraphs with parameters previously unknown, and generalizes to rainbow (q,p)-colorings with new combinatorial theorems.

Findings

NP-hardness of 2-coloring for rainbow (k - 2√k)-colorable hypergraphs

NP-hardness of c-coloring for rainbow (q,p)-colorable hypergraphs with specific parameters

Use of topological and combinatorial theorems in proofs

Abstract

A rainbow $q$-coloring of a $k$-uniform hypergraph is a $q$-coloring of the vertex set such that every hyperedge contains all $q$ colors. We prove that given a rainbow $(k - 2\lfloor \sqrt{k}\rfloor)$-colorable $k$-uniform hypergraph, it is NP-hard to find a normal $2$-coloring. Previously, this was only known for rainbow $\lfloor k/2 \rfloor$-colorable hypergraphs (Guruswami and Lee, SODA 2015). We also study a generalization which we call rainbow $(q, p)$-coloring, defined as a coloring using $q$ colors such that every hyperedge contains at least $p$ colors. We prove that given a rainbow $(k - \lfloor \sqrt{kc} \rfloor, k- \lfloor3\sqrt{kc} \rfloor)$-colorable $k$ uniform hypergraph, it is NP-hard to find a normal $c$-coloring for any $c = o(k)$. The proof of our second result relies on two combinatorial theorems. One of the theorems was proved by Sarkaria (J. Comb. Theory. 1990) using topological methods and the other theorem we prove using a generalized Borsuk-Ulam theorem.