### 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 disjunctive normal form (DNF), and Min-Circuit (also called the minimum circuit size problem (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 [Some NP-Complete Set Covering Problems, manuscript, 1979], 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 quasi-polynomial 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 turn to the question of approximating circuit size for slightly more general classes of circuits. DNF formulas are depth-two circuits of AND and OR gates. Depth-d circuits are denoted by AC^{0}
_{d}. We show that it is hard to approximate the size of AC^{0}
_{d}. circuits (for large enough d) under cryptographic assumptions.

Original language | English (US) |
---|---|

Pages (from-to) | 63-84 |

Number of pages | 22 |

Journal | SIAM Journal on Computing |

Volume | 38 |

Issue number | 1 |

DOIs | |

State | Published - 2008 |

### Fingerprint

### Keywords

- Approximation algorithms
- Complexity theory
- Machine learning theory
- Truth table minimization

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science(all)

### Cite this

^{0}circuits given a truth table.

*SIAM Journal on Computing*,

*38*(1), 63-84. https://doi.org/10.1137/060664537

**Minimizing disjunctive normal form formulas and AC ^{0} circuits given a truth table.** / Allender, Eric; Hellerstein, Lisa; Mccabe, Paul; Pitassi, Toniann; Saks, Michael.

Research output: Contribution to journal › Article

^{0}circuits given a truth table',

*SIAM Journal on Computing*, vol. 38, no. 1, pp. 63-84. https://doi.org/10.1137/060664537

^{0}circuits given a truth table. SIAM Journal on Computing. 2008;38(1):63-84. https://doi.org/10.1137/060664537

}

TY - JOUR

T1 - Minimizing disjunctive normal form formulas and AC0 circuits given a truth table

AU - Allender, Eric

AU - Hellerstein, Lisa

AU - Mccabe, Paul

AU - Pitassi, Toniann

AU - Saks, Michael

PY - 2008

Y1 - 2008

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 disjunctive normal form (DNF), and Min-Circuit (also called the minimum circuit size problem (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 [Some NP-Complete Set Covering Problems, manuscript, 1979], 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 quasi-polynomial 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 turn to the question of approximating circuit size for slightly more general classes of circuits. DNF formulas are depth-two circuits of AND and OR gates. Depth-d circuits are denoted by AC0 d. We show that it is hard to approximate the size of AC0 d. circuits (for large enough d) 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 disjunctive normal form (DNF), and Min-Circuit (also called the minimum circuit size problem (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 [Some NP-Complete Set Covering Problems, manuscript, 1979], 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 quasi-polynomial 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 turn to the question of approximating circuit size for slightly more general classes of circuits. DNF formulas are depth-two circuits of AND and OR gates. Depth-d circuits are denoted by AC0 d. We show that it is hard to approximate the size of AC0 d. circuits (for large enough d) under cryptographic assumptions.

KW - Approximation algorithms

KW - Complexity theory

KW - Machine learning theory

KW - Truth table minimization

UR - http://www.scopus.com/inward/record.url?scp=55249100776&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=55249100776&partnerID=8YFLogxK

U2 - 10.1137/060664537

DO - 10.1137/060664537

M3 - Article

AN - SCOPUS:55249100776

VL - 38

SP - 63

EP - 84

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 1

ER -