### 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 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### 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 |

### ASJC Scopus subject areas

- Computer Science(all)

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## Cite this

Bansal, N., & Khot, S. (2009). Optimal long code test with one free bit. In

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