Conditional hardness for approximate coloring

Irit Dinur, Elchanan Mossel, Oded Regev

Research output: Contribution to journalArticle

Abstract

We study the AprxColoring(q, Q) problem: Given a graph G, decide whether x(G)≤ q or x(G) ≥ Q. We present hardness results for this problem for any constants 3 ≤ q <Q. For q ≥4, our result is based on Khot's 2-to-1 label cover, which is conjectured to be NP-hard [S. Khot, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767-775]. For q = 3, we base our hardness result on a certain "⋉-shaped" variant of his conjecture. Previously no hardness result was known for q = 3 and Q ≥ 6. At the heart of our proof are tight bounds on generalized noise-stability quantities, which extend the recent work of Mossel, O'Donnell, and Oleszkiewicz ["Noise stability of functions with low influences: Invariance and optimality, " Ann. of Math. (2), to appear] and should have wider applicability.

Original languageEnglish (US)
Pages (from-to)843-873
Number of pages31
JournalSIAM Journal on Computing
Volume39
Issue number3
DOIs
StatePublished - 2009

Fingerprint

Coloring
Hardness
Colouring
Invariance
Labels
Annual
Optimality
NP-complete problem
Cover
Computing
Graph in graph theory

Keywords

  • Graph coloring
  • Hardness of approximation
  • Unique games

ASJC Scopus subject areas

  • Mathematics(all)
  • Computer Science(all)

Cite this

Conditional hardness for approximate coloring. / Dinur, Irit; Mossel, Elchanan; Regev, Oded.

In: SIAM Journal on Computing, Vol. 39, No. 3, 2009, p. 843-873.

Research output: Contribution to journalArticle

Dinur, Irit ; Mossel, Elchanan ; Regev, Oded. / Conditional hardness for approximate coloring. In: SIAM Journal on Computing. 2009 ; Vol. 39, No. 3. pp. 843-873.
@article{0975272a8cbf43b09153021915fd6360,
title = "Conditional hardness for approximate coloring",
abstract = "We study the AprxColoring(q, Q) problem: Given a graph G, decide whether x(G)≤ q or x(G) ≥ Q. We present hardness results for this problem for any constants 3 ≤ q <Q. For q ≥4, our result is based on Khot's 2-to-1 label cover, which is conjectured to be NP-hard [S. Khot, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767-775]. For q = 3, we base our hardness result on a certain {"}⋉-shaped{"} variant of his conjecture. Previously no hardness result was known for q = 3 and Q ≥ 6. At the heart of our proof are tight bounds on generalized noise-stability quantities, which extend the recent work of Mossel, O'Donnell, and Oleszkiewicz [{"}Noise stability of functions with low influences: Invariance and optimality, {"} Ann. of Math. (2), to appear] and should have wider applicability.",
keywords = "Graph coloring, Hardness of approximation, Unique games",
author = "Irit Dinur and Elchanan Mossel and Oded Regev",
year = "2009",
doi = "10.1137/07068062X",
language = "English (US)",
volume = "39",
pages = "843--873",
journal = "SIAM Journal on Computing",
issn = "0097-5397",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "3",

}

TY - JOUR

T1 - Conditional hardness for approximate coloring

AU - Dinur, Irit

AU - Mossel, Elchanan

AU - Regev, Oded

PY - 2009

Y1 - 2009

N2 - We study the AprxColoring(q, Q) problem: Given a graph G, decide whether x(G)≤ q or x(G) ≥ Q. We present hardness results for this problem for any constants 3 ≤ q <Q. For q ≥4, our result is based on Khot's 2-to-1 label cover, which is conjectured to be NP-hard [S. Khot, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767-775]. For q = 3, we base our hardness result on a certain "⋉-shaped" variant of his conjecture. Previously no hardness result was known for q = 3 and Q ≥ 6. At the heart of our proof are tight bounds on generalized noise-stability quantities, which extend the recent work of Mossel, O'Donnell, and Oleszkiewicz ["Noise stability of functions with low influences: Invariance and optimality, " Ann. of Math. (2), to appear] and should have wider applicability.

AB - We study the AprxColoring(q, Q) problem: Given a graph G, decide whether x(G)≤ q or x(G) ≥ Q. We present hardness results for this problem for any constants 3 ≤ q <Q. For q ≥4, our result is based on Khot's 2-to-1 label cover, which is conjectured to be NP-hard [S. Khot, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767-775]. For q = 3, we base our hardness result on a certain "⋉-shaped" variant of his conjecture. Previously no hardness result was known for q = 3 and Q ≥ 6. At the heart of our proof are tight bounds on generalized noise-stability quantities, which extend the recent work of Mossel, O'Donnell, and Oleszkiewicz ["Noise stability of functions with low influences: Invariance and optimality, " Ann. of Math. (2), to appear] and should have wider applicability.

KW - Graph coloring

KW - Hardness of approximation

KW - Unique games

UR - http://www.scopus.com/inward/record.url?scp=69049114323&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69049114323&partnerID=8YFLogxK

U2 - 10.1137/07068062X

DO - 10.1137/07068062X

M3 - Article

AN - SCOPUS:69049114323

VL - 39

SP - 843

EP - 873

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 3

ER -