Khot-Saket超图着色硬度约简中的一致性约简
Reducing uniformity in Khot-Saket hypergraph coloring hardness reductions
摘要：
在最近的结果中，Khot和Saket [FOCS 2014]证明了用$2^{(\log n)^{Ω(1)}}$颜色着色2-可着色的12-均匀超图的准NP-硬度。使用一种新的外部PCP验证器证明了这一结果，该外部PCP验证器具有很强的可靠性保证。在本文中，我们表明，我们可以减少他们的结果的arity，通过修改他们的12-查询内部验证到一个8-查询内部验证的基础上超图着色硬度约简Guruswami等。[STOC 2014]。更确切地说，我们证明了以下问题的拟NP-困难的n-顶点超图。- 用$2^{(\log n)^{Ω(1)}}$色着色一个2-可着色的8-均匀超图. - 用$2^{(\log n)^{Ω(1)}}$色着色4-可着色4-均匀超图。
英文摘要：
In a recent result, Khot and Saket [FOCS 2014] proved the quasi-NP-hardness of coloring a 2-colorable 12-uniform hypergraph with $2^{(\log n)^{Ω(1)}}$ 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 $2^{(\log n)^{Ω(1)}}$ colors. - Coloring a 4-colorable 4-uniform hypergraph with $2^{(\log n)^{Ω(1)}}$ colors.
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