Approximation of the size of the union

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-1/2ⁿ) is defined for any k and n such that n - k is at most a constant.