Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions

Abstract.

Let $A_{1}, \ldots, A_{n}$ be events in a probability space. The approximate inclusion-exclusion problem, due to Linial and Nisan (1990), is to estimate ${\bf P} [A_{1} \cup \ldots \cup A_{n}]$ given ${\bf P} [\bigcap_{i\in S} A_{i}]$ for |S| ≤ k. Kahn et al. (1996) solved this problem optimally for each k. We study the following more general question: estimate ${\bf P} [f(A_{1}, \ldots, A_{n})]$ given ${\bf P} [\bigcap_{i\in S} A_{i}]$ for |S| ≤ k, where f : {0, 1}ⁿ → {0, 1} is a given symmetric function. We solve this general problem for every f and k, giving an algorithm that runs in polynomial time and achieves an approximation error that is essentially optimal. We prove this optimal error to be $2^{-\tilde\Theta(k^{2}/n)}$ for k above a certain threshold, and $\Theta(1)$ otherwise.

As part of our solution, we analyze, for every nonconstant symmetric f : {0, 1}ⁿ → {0, 1} and every $\epsilon \in [2^{-n}, 1/3]$, the least degree ${\rm deg}_{\epsilon}(f)$ of a polynomial that approximates f pointwise within $\epsilon$. We show that ${\rm deg}_{\epsilon}(f) = \tilde\Theta({\rm deg}_{1/3}(f) + \sqrt{n {\rm log}(1/\epsilon))}$, where deg_1/3(f) is well-known for each f. Previously, the answer for vanishing $\epsilon$ was known only for f = OR. We construct the approximating polynomial explicitly for all f and $\epsilon$.