Simplifying inclusion-exclusion formulas

TL;DR

This paper presents a method to simplify the classical inclusion-exclusion formulas for arbitrary set families, reducing the number of terms from exponential to quasi-polynomial size, and analyzes the complexity and limitations of such simplifications.

Contribution

It provides an upper bound on the size of simplified inclusion-exclusion formulas for any set family and demonstrates the computational feasibility of constructing these formulas.

Findings

Simplified formulas have $m^{O( ext{log}^2 n)}$ terms.

Such formulas can be computed efficiently in expected polynomial time.

There are families where any valid formula must have large coefficients, indicating inherent complexity.

Abstract

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 $\Omega(m^{2-\varepsilon})$.