2(logN) Hardness for Hypergraph Coloring

We show that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with 2(logN) colors, where N is the number of vertices. There has been much focus on hardness of hypergraph coloring recently. In [17], Guruswami, Håstad, Harsha, Srinivasan and Varma showed that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with 22 √ log logN) 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 [12]. Using a different approach in [30], Khot and Saket constructed a new variant of Label-Cover, and composed it withQuadratic-Code to show quasi-NP-hardness of coloring 2-colorable 12-uniform hypergraphs with 2(logN)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 [39]. 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 a 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 [12] in the context of analyzing probabilistically checkable proof systems.