## Abstract

For circuit classes R, the fundamental computational problem Min-R asks for the minimum R-size of a Boolean function presented as a truth table. Prominent examples of this problem include Min-DNF, which asks whether a given Boolean function presented as a truth table has a k-term DNF, and Min-Circuit (also called MCSP), which asks whether a Boolean function presented as a truth table has a size k Boolean circuit. We present a new reduction proving that Min-DNF is NP-complete. It is significantly simpler than the known reduction of Masek [31], which is from Circuit-SAT. We then give a more complex reduction, yielding the result that Min-DNF cannot be approximated to within a factor smaller than (log N)^{γ}, for some constant γ > 0, assuming that NP is not contained in quasipolynomial time. The standard greedy algorithm for Set Cover is often used in practice to approximate Min-DNF. The question of whether Min-DNF can be approximated to within a factor of o(log N) remains open, but we construct an instance of Min-DNF on which the solution produced by the greedy algorithm is Ω(log N) larger than optimal. Finally, we extend known hardness results for Min-TC_{d}^{0} to obtain new hardness results for Min-AC_{d}^{0}, under cryptographic assumptions.

Original language | English (US) |
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Title of host publication | Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006 |

Pages | 237-251 |

Number of pages | 15 |

DOIs | |

State | Published - 2006 |

Event | 21st Annual IEEE Conference on Computational Complexity, CCC 2006 - Prague, Czech Republic Duration: Jul 16 2006 → Jul 20 2006 |

### Publication series

Name | Proceedings of the Annual IEEE Conference on Computational Complexity |
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Volume | 2006 |

ISSN (Print) | 1093-0159 |

### Other

Other | 21st Annual IEEE Conference on Computational Complexity, CCC 2006 |
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Country/Territory | Czech Republic |

City | Prague |

Period | 7/16/06 → 7/20/06 |

## All Science Journal Classification (ASJC) codes

- Software
- Theoretical Computer Science
- Computational Mathematics

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