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Simplifying inclusion — exclusion formulas

  • Xavier Goaoc
  • Jiří Matoušek
  • Pavel Paták
  • Zuzana Safernová
  • Martin Tancer
Conference paper
Part of the CRM Series book series (PSNS, 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.

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Copyright information

© Scuola Normale Superiore Pisa 2013

Authors and Affiliations

  • Xavier Goaoc
    • 1
    • 2
    • 3
  • Jiří Matoušek
    • 4
    • 5
  • Pavel Paták
    • 6
  • Zuzana Safernová
    • 4
  • Martin Tancer
    • 4
  1. 1.Université de LorraineFrance
  2. 2.CNRSFrance
  3. 3.InriaFrance
  4. 4.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic
  5. 5.Institute of Theoretical Computer ScienceETH ZurichZurichSwitzerland
  6. 6.Department of AlgebraCharles UniversityPraha 8Czech Republic

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