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Simplifying Inclusion–Exclusion Formulas
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- Journal:
- Combinatorics, Probability and Computing / Volume 24 / Issue 2 / March 2015
- Published online by Cambridge University Press:
- 14 October 2014, pp. 438-456
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- Article
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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
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
$\[\mu(F_1\cup F_2\cup\cdots\cup F_n)=\sum_{I:\emptyset\ne I\subseteq[n]} (-1)^{|I|+1}\mu\biggl(\bigcap_{i\in I} F_i\biggr),\]$$\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).
