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Simplifying inclusion-exclusion formulas

Identifieur interne : 001039 ( Main/Exploration ); précédent : 001038; suivant : 001040

Simplifying inclusion-exclusion formulas

Auteurs : Xavier Goaoc [France] ; Ji Matoušek [République tchèque] ; Pavel Paták [République tchèque] ; Zuzana Safernová [République tchèque] ; Martin Tancer [République tchèque]

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RBID : Hal:hal-00764182

Abstract

Let F = {F_1, F_2,..., F_n} be a family of n sets on a ground set X, such as a family of balls in R^d. For every finite measure \mu on X, such that the sets of F are measurable, the classical inclusion-exclusion formula asserts that \mu(F_1 \cup F_2 \cup \bullet \bullet \bullet \cup F_n) = \sum_{I:{\O} \neq I\subseteq[n]} (-1)^{|I|+1} \mu(\cap_{i\inI} F_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 the apparently first 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((log n)^2) terms and with \pm1 coefficients, and that such a formula can be computed in m^O((log n)^2) expected time. We also construct systems of n sets on n points for which every valid inclusion-exclusion formula has the sum of absolute values of the coefficients at least \Omega(n^{3/2}).

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</hal:affiliation>
<country>République tchèque</country>
</affiliation>
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<front>
<div type="abstract" xml:lang="en">Let F = {F_1, F_2,..., F_n} be a family of n sets on a ground set X, such as a family of balls in R^d. For every finite measure \mu on X, such that the sets of F are measurable, the classical inclusion-exclusion formula asserts that \mu(F_1 \cup F_2 \cup \bullet \bullet \bullet \cup F_n) = \sum_{I:{\O} \neq I\subseteq[n]} (-1)^{|I|+1} \mu(\cap_{i\inI} F_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 the apparently first 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((log n)^2) terms and with \pm1 coefficients, and that such a formula can be computed in m^O((log n)^2) expected time. We also construct systems of n sets on n points for which every valid inclusion-exclusion formula has the sum of absolute values of the coefficients at least \Omega(n^{3/2}).</div>
</front>
</TEI>
<affiliations>
<list>
<country>
<li>France</li>
<li>République tchèque</li>
</country>
<region>
<li>Grand Est</li>
<li>Lorraine (région)</li>
</region>
<settlement>
<li>Metz</li>
<li>Nancy</li>
</settlement>
<orgName>
<li>Université de Lorraine</li>
</orgName>
</list>
<tree>
<country name="France">
<region name="Grand Est">
<name sortKey="Goaoc, Xavier" sort="Goaoc, Xavier" uniqKey="Goaoc X" first="Xavier" last="Goaoc">Xavier Goaoc</name>
</region>
</country>
<country name="République tchèque">
<noRegion>
<name sortKey="Matousek, Ji" sort="Matousek, Ji" uniqKey="Matousek J" first="Ji" last="Matoušek">Ji Matoušek</name>
</noRegion>
<name sortKey="Patak, Pavel" sort="Patak, Pavel" uniqKey="Patak P" first="Pavel" last="Paták">Pavel Paták</name>
<name sortKey="Safernova, Zuzana" sort="Safernova, Zuzana" uniqKey="Safernova Z" first="Zuzana" last="Safernová">Zuzana Safernová</name>
<name sortKey="Tancer, Martin" sort="Tancer, Martin" uniqKey="Tancer M" first="Martin" last="Tancer">Martin Tancer</name>
</country>
</tree>
</affiliations>
</record>

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