NP-Hardness of Almost Coloring Almost 3-Colorable Graphs

Yahli Hecht; Dor Minzer; Muli Safra

doi:10.4230/LIPIcs.APPROX/RANDOM.2023.51

NP-Hardness of Almost Coloring Almost 3-Colorable Graphs

Yahli Hecht^*, Dor Minzer^*, Muli Safra^*

^*Corresponding author for this work

School of Computer Science

Massachusetts Institute of Technology

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

2 Scopus citations

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. [16], who showed a weaker result wherein in the “yes case” the graph is (4, ε) almost colorable.

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023
Editors	Nicole Megow, Adam Smith
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959772969
DOIs	https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.51
State	Published - Sep 2023
Event	26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023 - Atlanta, United States Duration: 11 Sep 2023 → 13 Sep 2023

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	275
ISSN (Print)	1868-8969

Conference

Conference	26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023
Country/Territory	United States
City	Atlanta
Period	11/09/23 → 13/09/23

Funding

Funders	Funder number
National Science Foundation	2227876, 2239160
Bloom's Syndrome Foundation
Horizon 2020 Framework Programme	835152
Iowa Science Foundation
European Commission

Keywords

PCP, Hardness of approximation

Access to Document

10.4230/LIPIcs.APPROX/RANDOM.2023.51

Cite this

Hecht, Y., Minzer, D., & Safra, M. (2023). NP-Hardness of Almost Coloring Almost 3-Colorable Graphs. In N. Megow, & A. Smith (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023 Article 51 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 275). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.51

Hecht, Yahli ; Minzer, Dor ; Safra, Muli. / NP-Hardness of Almost Coloring Almost 3-Colorable Graphs. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023. editor / Nicole Megow ; Adam Smith. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2023. (Leibniz International Proceedings in Informatics, LIPIcs).

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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. [16], who showed a weaker result wherein in the “yes case” the graph is (4, ε) almost colorable.",

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Hecht, Y, Minzer, D & Safra, M 2023, NP-Hardness of Almost Coloring Almost 3-Colorable Graphs. in N Megow & A Smith (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023., 51, Leibniz International Proceedings in Informatics, LIPIcs, vol. 275, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023, Atlanta, United States, 11/09/23. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.51

NP-Hardness of Almost Coloring Almost 3-Colorable Graphs. / Hecht, Yahli; Minzer, Dor; Safra, Muli.
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023. ed. / Nicole Megow; Adam Smith. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2023. 51 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 275).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Hecht Y, Minzer D, Safra M. NP-Hardness of Almost Coloring Almost 3-Colorable Graphs. In Megow N, Smith A, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2023. 51. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.APPROX/RANDOM.2023.51

NP-Hardness of Almost Coloring Almost 3-Colorable Graphs

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