### 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/q^{k+1} vertices. This substantially improves upon the work of Dinur et al. [DKPS] who gave a corresponding bound of N/q^{2}. Our result implies that for any positive integer k, given a graph that has an independent set of ≈ (2^{k} + 1)^{-1} fraction of vertices, it is NP-hard to find an independent set of (2^{k} + 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 language | English (US) |
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Title of host publication | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

Pages | 380-389 |

Number of pages | 10 |

DOIs | |

State | Published - 2012 |

Event | 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States Duration: Oct 20 2012 → Oct 23 2012 |

### Other

Other | 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 |
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Country | United States |

City | New Brunswick, NJ |

Period | 10/20/12 → 10/23/12 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*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.

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

*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

}

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

AN - SCOPUS:84871969324

SP - 380

EP - 389

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

ER -