ECCC-Report TR15-062https://eccc.weizmann.ac.il/report/2015/062Comments and Revisions published for TR15-062en-usMon, 19 Oct 2015 06:51:19 +0300
Revision 2
| $2^{(\log N)^{1/10-o(1)}}$ Hardness for Hypergraph Coloring |
Sangxia Huang
https://eccc.weizmann.ac.il/report/2015/062#revision2We show that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with $2^{(\log N)^{1/10-o(1)}}$ colors, where $N$ is the number of vertices. There has been much focus on hardness of hypergraph coloring recently. Guruswami, H{\aa}stad, Harsha, Srinivasan and Varma showed that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with $2^{2^{\Omega(\sqrt{\log\log N})}}$ colors. Their result is obtained by composing standard Label Cover with an inner-verifier based on Low Degree Long Code, using Reed-Muller code testing results by Dinur and Guruswami. Using a different approach, Khot and Saket constructed a new variant of Label Cover, and composed it with Quadratic Code to show quasi-NP-hardness of coloring 2-colorable 12-uniform hypergraphs with $2^{(\log N)^c}$ colors, for some $c$ around 1/20. Their construction of Label Cover is based on a new notion of superposition complexity for CSP instances. The composition with inner-verifier was subsequently improved by Varma, giving the same hardness result for
8-uniform hypergraphs.
Our construction uses both Quadratic Code and Low Degree Long Code, and builds upon the work by Khot and Saket. We present a different approach to construct CSP instances with superposition hardness by observing that when the number of assignments is odd, satisfying a constraint in superposition is the same as "odd-covering" the constraint. We employ Low Degree Long Code in order to keep the construction efficient. In the analysis, we also adapt and generalize one of the key theorems by Dinur and Guruswami in the context of analyzing probabilistically checkable proof systems.Mon, 19 Oct 2015 06:51:19 +0300https://eccc.weizmann.ac.il/report/2015/062#revision2
Revision 1
| $2^{(\log N)^{1/8-o(1)}}$ Hardness for Hypergraph Coloring |
Sangxia Huang
https://eccc.weizmann.ac.il/report/2015/062#revision1We show that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with $2^{(\log N)^{1/8-o(1)}}$ colors, where $N$ is the number of vertices. There has been much focus on hardness of hypergraph coloring recently. Guruswami, H{\aa}stad, Harsha, Srinivasan and Varma showed that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with $2^{2^{\Omega(\sqrt{\log\log N})}}$ colors. Their result is obtained by composing standard Label Cover with an inner-verifier based on Low Degree Long Code, using Reed-Muller code testing results by Dinur and Guruswami. Using a different approach, Khot and Saket constructed a new variant of Label Cover, and composed it with Quadratic Code to show quasi-NP-hardness of coloring 2-colorable 12-uniform hypergraphs with $2^{(\log N)^c}$ colors, for some $c$ around 1/20. Their construction of Label Cover is based on a new notion of superposition complexity for CSP instances. The composition with inner-verifier was subsequently improved by Varma, giving the same hardness result for
8-uniform hypergraphs.
Our construction uses both Quadratic Code and Low Degree Long Code, and builds upon the work by Khot and Saket. We present a different approach to construct CSP instances with superposition hardness by observing that when the number of assignments is odd, satisfying a constraint in superposition is the same as "odd-covering" the constraint. We employ Low Degree Long Code in order to keep the construction efficient. In the analysis, we also adapt and generalize one of the key theorems by Dinur and Guruswami in the context of analyzing probabilistically checkable proof systems.Sun, 27 Sep 2015 22:21:00 +0300https://eccc.weizmann.ac.il/report/2015/062#revision1
Paper TR15-062
| $2^{(\log N)^{1/4-o(1)}}$ Hardness for Hypergraph Coloring |
Sangxia Huang
https://eccc.weizmann.ac.il/report/2015/062We show that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with $2^{(\log N)^{1/4-o(1)}}$ colors, where $N$ is the number of vertices. There has been much focus on hardness of hypergraph coloring recently. Guruswami, H{\aa}stad, Harsha, Srinivasan and Varma showed that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with $2^{2^{\Omega(\sqrt{\log\log N})}}$ colors. Their result is obtained by composing standard Label Cover with an inner-verifier based on Low Degree Long Code, using Reed-Muller code testing results by Dinur and Guruswami. Using a different approach, Khot and Saket constructed a new variant of Label Cover, and composed it with Quadratic Code to show quasi-NP-hardness of coloring 2-colorable 12-uniform hypergraphs with $2^{(\log N)^c}$ colors, for some $c$ around 1/20. Their construction of Label Cover is based on a new notion of superposition complexity for CSP instances. The composition with inner-verifier was subsequently improved by Varma, giving the same hardness result for
8-uniform hypergraphs.
Our construction uses both Quadratic Code and Low Degree Long Code, and builds upon the work by Khot and Saket. We present a different approach to construct CSP instances with superposition hardness by observing that when the number of assignments is odd, satisfying a constraint in superposition is the same as "odd-covering" the constraint. We employ Low Degree Long Code in order to keep the construction efficient. In the analysis, we also adapt and generalize one of the key theorems by Dinur and Guruswami in the context of analyzing probabilistically checkable proof systems.Wed, 15 Apr 2015 16:57:49 +0300https://eccc.weizmann.ac.il/report/2015/062