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arXiv · 1311.7407

Super-polylogarithmic hypergraph coloring hardness via low-degree long codes

Venkatesan Guruswami·Johan Hastad·Prahladh Harsha·Srikanth Srinivasan·Girish Varma
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Abstract

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the 'short code' of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results. In particular, we prove quasi-NP-hardness of the following problems on $n$-vertex hyper-graphs: * Coloring a 2-colorable 8-uniform hypergraph with $2^{2^{Ω(\sqrt{\log\log n})}}$ colors. * Coloring a 4-colorable 4-uniform hypergraph with $2^{2^{Ω(\sqrt{\log\log n})}}$ colors. * Coloring a 3-colorable 3-uniform hypergraph with $(\log n)^{Ω(1/\log\log\log n)}$ colors. In each of these cases, the hardness results obtained are (at least) exponentially stronger than what was previously known for the respective cases. In fact, prior to this result, polylog n colors was the strongest quantitative bound on the number of colors ruled out by inapproximability results for O(1)-colorable hypergraphs. The fundamental bottleneck in obtaining coloring inapproximability results using the low- degree long code was a multipartite structural restriction in the PCP construction of Dinur-Guruswami. We are able to get around this restriction by simulating the multipartite structure implicitly by querying just one partition (albeit requiring 8 queries), which yields our result for 2-colorable 8-uniform hypergraphs. The result for 4-colorable 4-uniform hypergraphs is obtained via a 'query doubling' method. For 3-colorable 3-uniform hypergraphs, we exploit the ternary domain to design a test with an additive (as opposed to multiplicative) noise function, and analyze its efficacy in killing high weight Fourier coefficients via the pseudorandom properties of an associated quadratic form.

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Venkatesan Guruswami, Johan Hastad, Prahladh Harsha, Srikanth Srinivasan, Girish Varma. 2013-11-28. Super-polylogarithmic hypergraph coloring hardness via low-degree long codes. https://doi.org/10.1137/140995520

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