### Abstract

Let k and n be integers with 1 ≤ h ≤ n; F(k, n) is the worst-case relative error generated in approximating the size of a finite system of n sets when the sizes of the intersections of all subsystems of at most k subsets are given. Linial et al. proposed a strategy to estimate F(k, n) and suggested that the strategy is applicable for the problem of counting the number of truth assignments for a DNF formula and the problem of calculating the permanent of a 0/1 matrix, both of which belong to the class of #P-complete problems. Although the time complexity increases as k increases, F(k, n) decreases their strategy. This paper gives elementary formulas that describe upper and lower bounds on F(k, n) for any k and n. The upper bound is better than the previous one when 2 ≤ n - k ≤ n/(8/log_{e} n). Furthermore, F(k, n) = 1 + Θ(√n^{n-k-1}/2^{n}) is defined for any k and n such that n - k is at most a constant.

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

Pages (from-to) | 11-23 |

Number of pages | 13 |

Journal | Systems and Computers in Japan |

Volume | 27 |

Issue number | 6 |

Publication status | Published - Jun 1996 |

### Fingerprint

### Keywords

- #P-complete
- Approximation algorithms
- Binomial coefficients
- Polynomial approximation

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Hardware and Architecture
- Information Systems
- Theoretical Computer Science

### Cite this

*Systems and Computers in Japan*,

*27*(6), 11-23.

**Approximation of the size of the union.** / Jinbo, Shuji; Maruoka, Akira.

Research output: Contribution to journal › Article

*Systems and Computers in Japan*, vol. 27, no. 6, pp. 11-23.

}

TY - JOUR

T1 - Approximation of the size of the union

AU - Jinbo, Shuji

AU - Maruoka, Akira

PY - 1996/6

Y1 - 1996/6

N2 - Let k and n be integers with 1 ≤ h ≤ n; F(k, n) is the worst-case relative error generated in approximating the size of a finite system of n sets when the sizes of the intersections of all subsystems of at most k subsets are given. Linial et al. proposed a strategy to estimate F(k, n) and suggested that the strategy is applicable for the problem of counting the number of truth assignments for a DNF formula and the problem of calculating the permanent of a 0/1 matrix, both of which belong to the class of #P-complete problems. Although the time complexity increases as k increases, F(k, n) decreases their strategy. This paper gives elementary formulas that describe upper and lower bounds on F(k, n) for any k and n. The upper bound is better than the previous one when 2 ≤ n - k ≤ n/(8/loge n). Furthermore, F(k, n) = 1 + Θ(√nn-k-1/2n) is defined for any k and n such that n - k is at most a constant.

AB - Let k and n be integers with 1 ≤ h ≤ n; F(k, n) is the worst-case relative error generated in approximating the size of a finite system of n sets when the sizes of the intersections of all subsystems of at most k subsets are given. Linial et al. proposed a strategy to estimate F(k, n) and suggested that the strategy is applicable for the problem of counting the number of truth assignments for a DNF formula and the problem of calculating the permanent of a 0/1 matrix, both of which belong to the class of #P-complete problems. Although the time complexity increases as k increases, F(k, n) decreases their strategy. This paper gives elementary formulas that describe upper and lower bounds on F(k, n) for any k and n. The upper bound is better than the previous one when 2 ≤ n - k ≤ n/(8/loge n). Furthermore, F(k, n) = 1 + Θ(√nn-k-1/2n) is defined for any k and n such that n - k is at most a constant.

KW - #P-complete

KW - Approximation algorithms

KW - Binomial coefficients

KW - Polynomial approximation

UR - http://www.scopus.com/inward/record.url?scp=0030173776&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030173776&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030173776

VL - 27

SP - 11

EP - 23

JO - Systems and Computers in Japan

JF - Systems and Computers in Japan

SN - 0882-1666

IS - 6

ER -