Approximate InclusionExclusion for Arbitrary Symmetric Functions
 Alexander A. Sherstov
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Let \(A_{1}, \ldots, A_{n}\) be events in a probability space. The approximate inclusionexclusion 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 wellknown 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 \(\epsilon\) .
 Title
 Approximate InclusionExclusion for Arbitrary Symmetric Functions
 Journal

computational complexity
Volume 18, Issue 2 , pp 219247
 Cover Date
 20090601
 DOI
 10.1007/s0003700902744
 Print ISSN
 10163328
 Online ISSN
 14208954
 Publisher
 BirkhäuserVerlag
 Additional Links
 Topics
 Keywords

 Approximate inclusion/exclusion
 approximate degree of Boolean functions
 approximation by polynomials
 03D15
 68Q17
 Industry Sectors
 Authors

 Alexander A. Sherstov ^{(1)}
 Author Affiliations

 1. Department of Computer Sciences, The University of Texas at Austin, 1 University Station C0500, Austin, TX, 787120233, USA