# Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions

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## 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 \(\epsilon\).

### Keywords.

Approximate inclusion/exclusion approximate degree of Boolean functions approximation by polynomials### Subject classification.

03D15 68Q17## Preview

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