, Volume 18, Issue 2, pp 219-247
Date: 12 Jun 2009

Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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 \(\epsilon\) .

Manuscript received 24 August 2008