computational complexity

, Volume 18, Issue 2, pp 219–247

# Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions

Article

## 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 deg1/3(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 $$\epsilon$$.

### Keywords.

Approximate inclusion/exclusion approximate degree of Boolean functions approximation by polynomials

03D15 68Q17