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/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.
| 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 |
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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
Approximation of the size of the union. / Jinbo, Shuji; Maruoka, Akira.
In: Systems and Computers in Japan, Vol. 27, No. 6, 06.1996, p. 11-23.Research output: Contribution to journal › Article
}
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
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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 -