### Abstract

The interior and exterior functions of a Boolean function f were introduced in Makino and Ibaraki (Discrete Appl. Math. 69 (1996) 209-231), as stability (or robustness) measures of the f. In this paper, we investigate the complexity of two problems α-INTERIOR and α-EXTERIOR, introduced therein. We first answer the question about the complexity of α-INTERIOR left open in Makino and Ibaraki (Discrete Appl. Math. 69 (1996) 209-231); it has no polynomial total time algorithm even if α is bounded by a constant, unless P = NP. However, for positive h-term DNF functions with h bounded by a constant, problems α-INTERIOR and α-EXTERIOR can be solved in (input) polynomial time and polynomial delay, respectively. Furthermore, for positive k-DNF functions, α-INTERIOR for two cases in which k = 1, and α and k are both bounded by a constant, can be solved in polynomial delay and in polynomial time, respectively.

Original language | English |
---|---|

Pages (from-to) | 417-436 |

Number of pages | 20 |

Journal | Discrete Applied Mathematics |

Volume | 130 |

Issue number | 3 |

DOIs | |

Publication status | Published - Aug 23 2003 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

*130*(3), 417-436. https://doi.org/10.1016/S0166-218X(02)00602-9

**Interior and exterior functions of positive Boolean functions.** / Makino, Kazuhisa; Ono, Hirotaka; Ibaraki, Toshihide.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 130, no. 3, pp. 417-436. https://doi.org/10.1016/S0166-218X(02)00602-9

}

TY - JOUR

T1 - Interior and exterior functions of positive Boolean functions

AU - Makino, Kazuhisa

AU - Ono, Hirotaka

AU - Ibaraki, Toshihide

PY - 2003/8/23

Y1 - 2003/8/23

N2 - The interior and exterior functions of a Boolean function f were introduced in Makino and Ibaraki (Discrete Appl. Math. 69 (1996) 209-231), as stability (or robustness) measures of the f. In this paper, we investigate the complexity of two problems α-INTERIOR and α-EXTERIOR, introduced therein. We first answer the question about the complexity of α-INTERIOR left open in Makino and Ibaraki (Discrete Appl. Math. 69 (1996) 209-231); it has no polynomial total time algorithm even if α is bounded by a constant, unless P = NP. However, for positive h-term DNF functions with h bounded by a constant, problems α-INTERIOR and α-EXTERIOR can be solved in (input) polynomial time and polynomial delay, respectively. Furthermore, for positive k-DNF functions, α-INTERIOR for two cases in which k = 1, and α and k are both bounded by a constant, can be solved in polynomial delay and in polynomial time, respectively.

AB - The interior and exterior functions of a Boolean function f were introduced in Makino and Ibaraki (Discrete Appl. Math. 69 (1996) 209-231), as stability (or robustness) measures of the f. In this paper, we investigate the complexity of two problems α-INTERIOR and α-EXTERIOR, introduced therein. We first answer the question about the complexity of α-INTERIOR left open in Makino and Ibaraki (Discrete Appl. Math. 69 (1996) 209-231); it has no polynomial total time algorithm even if α is bounded by a constant, unless P = NP. However, for positive h-term DNF functions with h bounded by a constant, problems α-INTERIOR and α-EXTERIOR can be solved in (input) polynomial time and polynomial delay, respectively. Furthermore, for positive k-DNF functions, α-INTERIOR for two cases in which k = 1, and α and k are both bounded by a constant, can be solved in polynomial delay and in polynomial time, respectively.

UR - http://www.scopus.com/inward/record.url?scp=0041386580&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041386580&partnerID=8YFLogxK

U2 - 10.1016/S0166-218X(02)00602-9

DO - 10.1016/S0166-218X(02)00602-9

M3 - Article

AN - SCOPUS:0041386580

VL - 130

SP - 417

EP - 436

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 3

ER -