NP-Hardness of Coloring 2-Colorable Hypergraph with Poly-Logarithmically Many Colors

Amey Bhangale, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
We give very short and simple proofs of the following statements: Given a 2-colorable 4-uniform hypergraph on n vertices, 1. It is NP-hard to color it with log δ n colors for some δ > 0. 2. It is quasi-NP-hard to color it with O log 1−o(1) n colors. In terms of NP-hardness, it improves the result of Guruswam, Håstad and Sudani [SIAM Journal on Computing, 2002], combined with Moshkovitz-Raz [Journal of the ACM, 2010], by an 'exponential' factor. The second result improves the result of Saket
more » ... ference on Computational Complexity (CCC), 2014] which shows quasi-NP-hardness of coloring a 2-colorable 4-uniform hypergraph with O (log γ n) colors for a sufficiently small constant 1 γ > 0. Our result is the first to show the NP-hardness of coloring a c-colorable k-uniform hypergraph with poly-logarithmically many colors, for any constants c ≥ 2 and k ≥ 3.
doi:10.4230/lipics.icalp.2018.15 dblp:conf/icalp/Bhangale18 fatcat:mgbf47aqozgf5dbcg6a33wvnba