### Abstract

For arbitrarily small constants ε, δ > 0, we present a long code test with one free bit, completeness 1 - ε and soundness δ. Using the test, we prove the following two inapproximability results: 1) Assuming the Unique Games Conjecture of Khot [15], given an n-vertex graph that has two disjoint independent sets of size (1/2 - ε)n each, it is NP-hard to find an independent set of size δn. 2) Assuming a (new) stronger version of the Unique Games Conjecture, the scheduling problem of minimizing weighted completion time with precedence constraints is inapprox-imable within factor 2 - ε.

Original language | English (US) |
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Title of host publication | Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009 |

Pages | 453-462 |

Number of pages | 10 |

DOIs | |

State | Published - 2009 |

Event | 50th Annual Symposium on Foundations of Computer Science, FOCS 2009 - Atlanta, GA, United States Duration: Oct 25 2009 → Oct 27 2009 |

### Other

Other | 50th Annual Symposium on Foundations of Computer Science, FOCS 2009 |
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Country | United States |

City | Atlanta, GA |

Period | 10/25/09 → 10/27/09 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009*(pp. 453-462). [5438609] https://doi.org/10.1109/FOCS.2009.23

**Optimal long code test with one free bit.** / Bansal, Nikhil; Khot, Subhash.

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

*Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009.*, 5438609, pp. 453-462, 50th Annual Symposium on Foundations of Computer Science, FOCS 2009, Atlanta, GA, United States, 10/25/09. https://doi.org/10.1109/FOCS.2009.23

}

TY - GEN

T1 - Optimal long code test with one free bit

AU - Bansal, Nikhil

AU - Khot, Subhash

PY - 2009

Y1 - 2009

N2 - For arbitrarily small constants ε, δ > 0, we present a long code test with one free bit, completeness 1 - ε and soundness δ. Using the test, we prove the following two inapproximability results: 1) Assuming the Unique Games Conjecture of Khot [15], given an n-vertex graph that has two disjoint independent sets of size (1/2 - ε)n each, it is NP-hard to find an independent set of size δn. 2) Assuming a (new) stronger version of the Unique Games Conjecture, the scheduling problem of minimizing weighted completion time with precedence constraints is inapprox-imable within factor 2 - ε.

AB - For arbitrarily small constants ε, δ > 0, we present a long code test with one free bit, completeness 1 - ε and soundness δ. Using the test, we prove the following two inapproximability results: 1) Assuming the Unique Games Conjecture of Khot [15], given an n-vertex graph that has two disjoint independent sets of size (1/2 - ε)n each, it is NP-hard to find an independent set of size δn. 2) Assuming a (new) stronger version of the Unique Games Conjecture, the scheduling problem of minimizing weighted completion time with precedence constraints is inapprox-imable within factor 2 - ε.

UR - http://www.scopus.com/inward/record.url?scp=77952368191&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952368191&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2009.23

DO - 10.1109/FOCS.2009.23

M3 - Conference contribution

SN - 9780769538501

SP - 453

EP - 462

BT - Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009

ER -