Super-Polylogarithmic Hypergraph Coloring Hardness via Low-Degree Long Codes

Abstract

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka the “short code” of Barak et al. [SIAM J. Comput., 44 (2015), pp. 1287--1324]) and the techniques proposed by Dinur and Guruswami [Israel J. Math., 209 (2015), pp. 611--649] to incorporate this code for inapproximability results. In particular, we prove quasi NP-hardness of the following problems on $n$-vertex hypergraphs: coloring a 2-colorable 8-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors; coloring a 4-colorable 4-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors; and coloring a 3-colorable 3-uniform hypergraph with $(\log n)^{\Omega(1/\log\log\log n)}$ colors. For the first two cases, the hardness results obtained are superpolynomial in what was previously known, and in the last case it is an exponential improvement. In fact, prior to this result, $(\log n)^{O(1)}$ colors was the strongest quantitative bound on the number of colors ruled out by inapproximability results for $O(1)$-colorable hypergraphs, and $(\log\log n)^{O(1)}$ for $O(1)$-colorable, 3-uniform hypergraphs.