Reducing uniformity in Khot-Saket hypergraph coloring hardness reductions

TL;DR

This paper improves the hardness results for coloring 2-colorable hypergraphs by reducing the arity from 12 to 8, demonstrating quasi-NP-hardness for coloring certain hypergraphs with exponentially many colors.

Contribution

It introduces a modified inner verifier that reduces the hypergraph arity from 12 to 8, strengthening the hardness of coloring problems.

Findings

Proves quasi-NP-hardness for coloring 8-uniform hypergraphs with many colors.

Establishes quasi-NP-hardness for 4-colorable 4-uniform hypergraphs.

Improves previous hypergraph coloring hardness results by reducing arity.

Abstract

In a recent result, Khot and Saket [FOCS 2014] proved the quasi-NP-hardness of coloring a 2-colorable 12-uniform hypergraph with $2^{(\log n)^{\Omega(1)}}$ colors. This result was proved using a novel outer PCP verifier which had a strong soundness guarantee. In this note, we show that we can reduce the arity of their result by modifying their 12-query inner verifier to an 8-query inner verifier based on the hypergraph coloring hardness reductions of Guruswami et. al. [STOC 2014]. More precisely, we prove quasi-NP-hardness of the following problems on n-vertex hypergraphs. - Coloring a 2-colorable 8-uniform hypergraph with $2^{(\log n)^{\Omega(1)}}$ colors. - Coloring a 4-colorable 4-uniform hypergraph with $2^{(\log n)^{\Omega(1)}}$ colors.