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Simplifying inclusion — exclusion formulas

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Part of the CRM Series book series (CRMSNS,volume 16)

Abstract

Let F = (F 1, F 2, …, F n) 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 F are measurable, the classical inclusion-exclusion formula asserts that μ (F 1 F 2 ∪ …∪ F n) = ∑I:ø≠⊆[n] (−1)¦I¦+1μ(∩i ∈IF i); 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 F. We provide an upper bound valid for an arbitrary F: we show that every system F of n sets with m nonempty 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.

Keywords

  • Simplicial Complex
  • Travel Salesman Problem
  • Venn Diagram
  • Expected Time
  • Bonferroni Inequality

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A visit of this author in Prague was partially supported from Grant GRADR Eurogiga GIG/11/E023

Supported by the ERC Advanced Grant No. 267165. Partially supported by the Charles University Grant GAUK 421511 and by Grant GRADR Eurogiga GIG/11/E023

Partially supported by the Charles University Grant GAUK 421511 and SVV-2012-265317

Supported by the ERC Advanced Grant No. 267165. Partially supported by the Charles University Grant GAUK 421511

Supported by the ERC Advanced Grant No. 267165. Partially supported by the Charles University Grant GAUK 421511

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© 2013 Scuola Normale Superiore Pisa

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Goaoc, X., Matoušek, J., Paták, P., Safernová, Z., Tancer, M. (2013). Simplifying inclusion — exclusion formulas. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_88

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