Simplifying inclusion-exclusion formulas

Let $\mathcal{F}=\{F_1,F_2, \ldots,F_n\}$ be a family of $n$ sets on a ground set $S$, such as a family of balls in $\mathbb{R}^d$. For every finite measure $\mu$ on $S$, such that the sets of $\mathcal{F}$ are measurable, the classical inclusion-exclusion formula asserts that $\mu(F_1\cup F_2\cup\cdots\cup F_n)=\sum_{I:\emptyset\ne I\subseteq[n]} (-1)^{|I|+1}\mu\Bigl(\bigcap_{i\in I} F_i\Bigr)$; 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 $m$ nonempty fields in the Venn diagram admits an inclusion-exclusion formula with $m^{O(\log^2n)}$ terms and with $\pm1$ coefficients, and that such a formula can be computed in $m^{O(\log^2n)}$ expected time. For every $\varepsilon>0$ 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 $\Omega(m^{2-\varepsilon})$.Comment: 17 pages, 3 figures/tables; improved lower bound in v

Numerical quadrature on the intersection of planar disks

We provide an algorithm that computes algebraic quadrature formulas with cardinality not exceeding the dimension of the exactness polynomial space, on the intersection of any number of planar disks with arbitrary radius. Applications arise for example in computational optics and in wireless networks analysis. By the inclusion-exclusion principle, we can also compute algebraic formulas for the union of a small number of disks. The algorithm is implemented in Matlab, via subperiodic trigonometric Gaussian quadrature and compression of discrete measures

Simplifying Inclusion-Exclusion Formulas

Let $\mathcal{F}$ = {F 1, F 2,..., Fn } be a family of n sets on a ground set S, such as a family of balls in ℝ d . 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 m O(log2 n) terms and with ±1 coefficients, and that such a formula can be computed in m O(log2 n) 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(m 2−ϵ

Simplifying Inclusion–Exclusion Formulas

ISSN:0963-5483ISSN:1469-216