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NP-Hardness of Almost Coloring Almost 3-Colorable Graphs

Authors Yahli Hecht , Dor Minzer , Muli Safra

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Author Details

Yahli Hecht
  • School of Computer Science, Tel Aviv University, Israel
Dor Minzer
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
Muli Safra
  • School of Computer Science, Tel Aviv University, Israel

Funding

  • Hecht, Yahli: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 835152).
  • Minzer, Dor: Supported by an Sloan Research Fellowship, NSF CCF award 2227876 and NSF CAREER award 2239160.
  • Safra, Muli: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 835152), and by an ISF grant and a BSF grant.

Cite As Get BibTex

Yahli Hecht, Dor Minzer, and Muli Safra. NP-Hardness of Almost Coloring Almost 3-Colorable Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.51

Abstract

A graph G = (V,E) is said to be (k,δ) almost colorable if there is a subset of vertices V' ⊆ V of size at least (1-δ)|V| such that the induced subgraph of G on V' is k-colorable. We prove that for all k, there exists δ > 0 such for all ε > 0, given a graph G it is NP-hard (under randomized reductions) to distinguish between:  
1) Yes case: G is (3,ε) almost colorable. 
2) No case: G is not (k,δ) almost colorable.  This improves upon an earlier result of Khot et al. [Irit Dinur et al., 2018], who showed a weaker result wherein in the "yes case" the graph is (4,ε) almost colorable.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • PCP
  • Hardness of approximation

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References

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