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)


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.


Simplicial Complex Travel Salesman Problem Venn Diagram Expected Time Bonferroni Inequality 
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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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