Inclusion-exclusion: exact and approximate (Q1375692)

Given \(n\) events if the probabilities of all \(\leq k\)-wise intersections are given the probability of the union can be calculated within an error \(\exp \bigl( - \Omega(k^2/n)\bigr)\). This is optimal. Given a Boolean formula \(F\) in DNF with \(m\) clauses, if it is known how many assignments satisfy any choice of at most \(\log n +1\) clauses, the number of assignments satisfying \(F\) can be calculated. The results are applied to the edge reconstruction problem, computing permanents, chromatic polynomials, and problems on VC dimensions.