Simplifying inclusion-exclusion formulas

Abstract

Let {F} = {F 1, F 2,.,Fn} be a family of n sets on a ground set S, such as a family of balls in d . For every finite measure μ on S, such that the sets of {F} are measurable, the classical inclusion-exclusion formula asserts that [EQUATION PRESENTED] that is, the measure of the union is expressed using measures of various intersections. The number of terms in this formula is exponential in n, and a significant amount of research, originating in applied areas, has been devoted to constructing simpler formulas for particular families \mathcal{F}. We provide an upper bound valid for an arbitrary {F}: we show that every system \mathcal{F} of n sets with m non-empty fields in the Venn diagram admits an inclusion-exclusion formula with mO(log2 n) terms and with ±1 coefficients, and that such a formula can be computed in mO(log2 n) expected time. For every Ï > 0 we also construct systems with Venn diagram of size m for which every valid inclusion-exclusion formula has the sum of absolute values of the coefficients at least Ω(m2-ε).