Simplifying Inclusion–Exclusion Formulas

Abstract

Let $\mathcal{F}$ = {F1, F2,. . ., Fn} be a family of n sets on a ground set S, such as a family of balls in ℝd. For every finite measure μ on S, such that the sets of $\mathcal{F}$ are measurable, the classical inclusion–exclusion formula asserts that $\[\mu(F_1\cup F_2\cup\cdots\cup F_n)=\sum_{I:\emptyset\ne I\subseteq[n]} (-1)^{|I|+1}\mu\biggl(\bigcap_{i\in I} F_i\biggr),\]$ 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 $\mathcal{F}$: 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(log2n) terms and with ±1 coefficients, and that such a formula can be computed in mO(log2n) expected time. For every ϵ > 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).