Hardness of finding independent sets in almost q-colorable graphs

Subhash Khot, Rishi Saket

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We show that for any ε > 0, and positive integers k and q such that q ≥= 2k + 1, given a graph on N vertices that has a q-colorable induced sub graph of (1 - ε)N vertices, it is NP-hard to find an independent set of N/qk+1 vertices. This substantially improves upon the work of Dinur et al. [DKPS] who gave a corresponding bound of N/q2. Our result implies that for any positive integer k, given a graph that has an independent set of ≈ (2k + 1)-1 fraction of vertices, it is NP-hard to find an independent set of (2k + 1)-(k+1) fraction of vertices. This improves on the previous work of Engebretsen and Holmerin [EH] who proved a gap of ≈ 2-k vs 2-(k 2), which is best possible using techniques (including those of [EH]) based on the query efficient PCP of Samorodnitsky and Trevisan [3].

Original languageEnglish (US)
Title of host publicationProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Pages380-389
Number of pages10
DOIs
StatePublished - 2012
Event53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States
Duration: Oct 20 2012Oct 23 2012

Other

Other53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012
CountryUnited States
CityNew Brunswick, NJ
Period10/20/1210/23/12

Fingerprint

Hardness

Keywords

  • Coloring
  • Graphs
  • Hardness
  • Independent-Set
  • PCP

ASJC Scopus subject areas

  • Computer Science(all)

Cite this

Khot, S., & Saket, R. (2012). Hardness of finding independent sets in almost q-colorable graphs. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS (pp. 380-389). [6375316] https://doi.org/10.1109/FOCS.2012.75

Hardness of finding independent sets in almost q-colorable graphs. / Khot, Subhash; Saket, Rishi.

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. 2012. p. 380-389 6375316.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Khot, S & Saket, R 2012, Hardness of finding independent sets in almost q-colorable graphs. in Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS., 6375316, pp. 380-389, 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, United States, 10/20/12. https://doi.org/10.1109/FOCS.2012.75
Khot S, Saket R. Hardness of finding independent sets in almost q-colorable graphs. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. 2012. p. 380-389. 6375316 https://doi.org/10.1109/FOCS.2012.75
Khot, Subhash ; Saket, Rishi. / Hardness of finding independent sets in almost q-colorable graphs. Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. 2012. pp. 380-389
@inproceedings{ceaa6fb5d5fe41ed8dd21cc694001f23,
title = "Hardness of finding independent sets in almost q-colorable graphs",
abstract = "We show that for any ε > 0, and positive integers k and q such that q ≥= 2k + 1, given a graph on N vertices that has a q-colorable induced sub graph of (1 - ε)N vertices, it is NP-hard to find an independent set of N/qk+1 vertices. This substantially improves upon the work of Dinur et al. [DKPS] who gave a corresponding bound of N/q2. Our result implies that for any positive integer k, given a graph that has an independent set of ≈ (2k + 1)-1 fraction of vertices, it is NP-hard to find an independent set of (2k + 1)-(k+1) fraction of vertices. This improves on the previous work of Engebretsen and Holmerin [EH] who proved a gap of ≈ 2-k vs 2-(k 2), which is best possible using techniques (including those of [EH]) based on the query efficient PCP of Samorodnitsky and Trevisan [3].",
keywords = "Coloring, Graphs, Hardness, Independent-Set, PCP",
author = "Subhash Khot and Rishi Saket",
year = "2012",
doi = "10.1109/FOCS.2012.75",
language = "English (US)",
pages = "380--389",
booktitle = "Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS",

}

TY - GEN

T1 - Hardness of finding independent sets in almost q-colorable graphs

AU - Khot, Subhash

AU - Saket, Rishi

PY - 2012

Y1 - 2012

N2 - We show that for any ε > 0, and positive integers k and q such that q ≥= 2k + 1, given a graph on N vertices that has a q-colorable induced sub graph of (1 - ε)N vertices, it is NP-hard to find an independent set of N/qk+1 vertices. This substantially improves upon the work of Dinur et al. [DKPS] who gave a corresponding bound of N/q2. Our result implies that for any positive integer k, given a graph that has an independent set of ≈ (2k + 1)-1 fraction of vertices, it is NP-hard to find an independent set of (2k + 1)-(k+1) fraction of vertices. This improves on the previous work of Engebretsen and Holmerin [EH] who proved a gap of ≈ 2-k vs 2-(k 2), which is best possible using techniques (including those of [EH]) based on the query efficient PCP of Samorodnitsky and Trevisan [3].

AB - We show that for any ε > 0, and positive integers k and q such that q ≥= 2k + 1, given a graph on N vertices that has a q-colorable induced sub graph of (1 - ε)N vertices, it is NP-hard to find an independent set of N/qk+1 vertices. This substantially improves upon the work of Dinur et al. [DKPS] who gave a corresponding bound of N/q2. Our result implies that for any positive integer k, given a graph that has an independent set of ≈ (2k + 1)-1 fraction of vertices, it is NP-hard to find an independent set of (2k + 1)-(k+1) fraction of vertices. This improves on the previous work of Engebretsen and Holmerin [EH] who proved a gap of ≈ 2-k vs 2-(k 2), which is best possible using techniques (including those of [EH]) based on the query efficient PCP of Samorodnitsky and Trevisan [3].

KW - Coloring

KW - Graphs

KW - Hardness

KW - Independent-Set

KW - PCP

UR - http://www.scopus.com/inward/record.url?scp=84871969324&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871969324&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2012.75

DO - 10.1109/FOCS.2012.75

M3 - Conference contribution

SP - 380

EP - 389

BT - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

ER -