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d-To-1 Hardness of Coloring 3-Colorable Graphs with O(1) Colors

Authors Venkatesan Guruswami, Sai Sandeep

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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • graph coloring
  • hardness of approximation

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Abstract

The d-to-1 conjecture of Khot asserts that it is NP-hard to satisfy an ε fraction of constraints of a satisfiable d-to-1 Label Cover instance, for arbitrarily small ε > 0. We prove that the d-to-1 conjecture for any fixed d implies the hardness of coloring a 3-colorable graph with C colors for arbitrarily large integers C.
Earlier, the hardness of O(1)-coloring a 4-colorable graphs is known under the 2-to-1 conjecture, which is the strongest in the family of d-to-1 conjectures, and the hardness for 3-colorable graphs is known under a certain "fish-shaped" variant of the 2-to-1 conjecture.

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Venkatesan Guruswami and Sai Sandeep. d-To-1 Hardness of Coloring 3-Colorable Graphs with O(1) Colors. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 62:1-62:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.62

Author Details

Venkatesan Guruswami
  • Carnegie Mellon University, Pittsburgh, PA, USA
Sai Sandeep
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

Research supported in part by NSF grants CCF-1422045 and CCF-1908125.

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