Approximate Inclusion - Exclusion for Arbitrary Symmetric Functions ∗

Let A 1 , A 2 ,. .. , A n be events in some probability space. The approximate inclusion-exclusion problem, due to Linial and Nisan (1990), is to estimate Pr[A 1 ∪· · ·∪A n ] given Pr[ i∈S A i ] for all |S | k. Kahn et al. (1996) solve this problem optimally for each k. We study the following more general question: given Pr[ i∈S A i ] for all |S | k, estimate Pr the number of events among A 1 ,. .. , A n that hold is in Z , where Z ⊆ {0, 1,. .. , n} is a given set. (In the Linial-Nisan problem, Z = {1,. .. , n}.) We solve this general problem for all Z 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 − ˜ Θ(k 2 /n) for k above a certain threshold, and Θ(1) otherwise. and every ∈ [1/2 n , 1/3], the least degree deg (D) of a polynomial that approximates D pointwise within. Namely, we show that deg (D) = ˜ Θ deg 1/3 (D) + n log(1/) , where deg 1/3 (D) is well-known for each D. Previously, the answer for vanishing was known only for D = OR (Kahn et al., 1996). We construct the approximating polynomial explicitly for every D and. Our proof departs considerably from Linial and Nisan (1990) and Kahn et al. (1996). Its key ingredient is the Approximation/Orthogonality Principle, a certain equivalence of approximation and orthogonality in a Euclidean space, recently proved by the author in the context of quantum lower bounds (Sherstov 2007). Our polynomial constructions feature new uses of the Chebyshev polynomials.