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

Author Amey Bhangale

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Amey Bhangale
  • Weizmann Institute of Science, Rehovot, Israel

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Amey Bhangale. NP-Hardness of Coloring 2-Colorable Hypergraph with Poly-Logarithmically Many Colors. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 15:1-15:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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^delta n colors for some delta>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 [Conference on Computational Complexity (CCC), 2014] which shows quasi-NP-hardness of coloring a 2-colorable 4-uniform hypergraph with O(log^gamma n) colors for a sufficiently small constant 1 >> gamma>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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Hypergraph coloring
  • Inapproximability
  • Schrijver graph


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