Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions

Let A1, . . . , An be events in a probability space. The approximate inclusion-exclusion problem, due to Linial and Nisan (1990), is to estimate P[A1 ∪ · · · ∪ An] given P[ ⋂ i∈S Ai] for |S | 6 k. Kahn et al. (1996) solved this problem optimally for each k. We study the following more general question: estimate P[ f (A1, . . . , An)] given P[ ⋂ i∈S Ai] for |S | 6 k, where f : {0, 1}n → {0, 1} is a given symmetric function. (In the Linial-Nisan problem, f = OR.) 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−Θ̃(k 2/n) for k above a certain threshold, and Θ(1) otherwise. As part of our solution, we analyze, for every nonconstant symmetric f : {0, 1}n → {0, 1} and every ∈ [2−n, 1/3], the least degree deg ( f ) of a polynomial that approximates f pointwise within . Namely, we show that deg ( f ) = Θ̃ ( deg1/3( f ) + √ n log(1/ ) ) , where deg1/3( f ) is well-known for each f . Previously, the answer for vanishing was known only for f = OR (Kahn et al., 1996; Buhrman et al., 1999). We construct the approximating polynomial explicitly for every f and . Our proof departs considerably from Linial and Nisan (1990) and Kahn et al. (1996). Its key ingredient is a classical result from approximation theory, which gives a certain equivalence of approximation and orthogonality in Euclidean space. Our polynomial constructions feature new uses of Chebyshev polynomials. ∗Department of Computer Sciences, The University of Texas at Austin, TX 78712 USA. Email: sherstov@cs.utexas.edu.