### 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 |

Volume | 2006 |

DOIs | |

State | Published - 2006 |

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

### Other

Other | 21st Annual IEEE Conference on Computational Complexity, CCC 2006 |
---|---|

Country | Czech Republic |

City | Prague |

Period | 7/16/06 → 7/20/06 |

### Fingerprint

### ASJC Scopus subject areas

- Computational Mathematics

### Cite this

*Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006*(Vol. 2006, pp. 237-251). [1663741] https://doi.org/10.1109/CCC.2006.27

**Inimizing DNF formulas and ACd
0 circuits given a truth table.** / Allender, Eric; Hellerstein, Lisa; McCabe, Paul; Pitassi, Toniann; Saks, Michael.

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

*Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006.*vol. 2006, 1663741, pp. 237-251, 21st Annual IEEE Conference on Computational Complexity, CCC 2006, Prague, Czech Republic, 7/16/06. https://doi.org/10.1109/CCC.2006.27

}

TY - GEN

T1 - Inimizing DNF formulas and ACd 0 circuits given a truth table

AU - Allender, Eric

AU - Hellerstein, Lisa

AU - McCabe, Paul

AU - Pitassi, Toniann

AU - Saks, Michael

PY - 2006

Y1 - 2006

N2 - 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-TCd 0 to obtain new hardness results for Min-ACd 0, under cryptographic assumptions.

AB - 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-TCd 0 to obtain new hardness results for Min-ACd 0, under cryptographic assumptions.

UR - http://www.scopus.com/inward/record.url?scp=34247522421&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247522421&partnerID=8YFLogxK

U2 - 10.1109/CCC.2006.27

DO - 10.1109/CCC.2006.27

M3 - Conference contribution

AN - SCOPUS:34247522421

SN - 0769525962

SN - 9780769525969

VL - 2006

SP - 237

EP - 251

BT - Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006

ER -