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

School of Computer Science

Tel Aviv University

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

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

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

Keywords

PCP, Hardness of approximation

All Science Journal Classification (ASJC) codes

Software

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.",

keywords = "PCP, Hardness of approximation",

author = "Yahli Hecht and Dor Minzer and Muli Safra",

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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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AU - Hecht, Yahli

AU - Minzer, Dor

AU - Safra, Muli

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N2 - 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.

AB - 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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M3 - منشور من مؤتمر

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023

A2 - Megow, Nicole

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PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

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Y2 - 11 September 2023 through 13 September 2023

ER -

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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