Improved Bonferroni Inequalities via Abstract Tubes: Inequalities and Identities of Inclusion-Exclusion Type

122 Pages·2003·0.7 MB·English

by  Klaus Dohmen (auth.)

Lecture Notes in Mathematics 1826
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
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Berlin
Heidelberg
NewYork
HongKong
London
Milan
Paris
Tokyo
Klaus Dohmen
Improved
Bonferroni Inequalities
via Abstract Tubes
Inequalities and Identities
of Inclusion-Exclusion Type
1 3
Author
KlausDohmen
DepartmentofMathematics
MittweidaUniversityofAppliedSciences
Technikumplatz17
09648Mittweida
Germany
e-mail:[email protected]
http://www.dohmen.htwm.de
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secondary:05A15,05B35,05C15,06A15,62N05,68M15,68R10,90B15,90B25
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Preface
This workis basedonmy habilitationthesis whichI preparedatBerlin’s Hum-
boldt-University while I was an assistant professor at the computer science de-
partment in the years 1994–2000. The material has been re-arranged, some
parts have been significantly shortened, while other parts have been expanded.
I am especially grateful to Professor Egmar R¨odel at Humboldt-University
for giving me the opportunity to work on this subject. Special thanks go to
the referees of my habilitation thesis as well as to the anonymous referees of
Springer-Verlagfortheirworthysuggestionsthatresultedinanimprovementof
themanuscript. ParticularthanksareowedtoProfessorDouglasShier(Clemson
University) for his valuable comments.
Finally, I would like to thank my wife Imke and my children Sophie, Emily
and Justus for their never-ending patience.
Mittweida, Germany Klaus Dohmen
July, 2003
Contents
1 Introduction and Overview 1
2 Preliminaries 5
2.1 Graphs and posets . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Bonferroni Inequalities via Abstract Tubes 9
3.1 An outline of abstract tube theory . . . . . . . . . . . . . . . . . 9
3.1.1 The notion of an abstract tube . . . . . . . . . . . . . . . 9
3.1.2 Two fundamental theorems on abstract tubes . . . . . . . 11
3.1.3 An importance sampling scheme . . . . . . . . . . . . . . 13
3.2 Examples of abstract tubes . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Abstract tubes for convex polyhedra . . . . . . . . . . . . 14
3.2.2 Abstract tubes for unions of balls and spherical caps . . . 15
4 Abstract Tubes via Closure and Kernel Operators 19
4.1 Abstract tubes via closure operators . . . . . . . . . . . . . . . . 19
4.1.1 Closure operators. . . . . . . . . . . . . . . . . . . . . . . 19
4.1.2 How closure operators lead to abstract tubes . . . . . . . 21
4.2 Abstract tubes via kernel operators . . . . . . . . . . . . . . . . . 25
4.2.1 Kernel operators . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.2 How kernel operators lead to abstract tubes . . . . . . . . 26
4.3 Alternative proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.1 Improved identities via closure operators. . . . . . . . . . 28
4.3.2 Improved inequalities via kernel operators . . . . . . . . . 33
4.3.3 Additional proofs of identities . . . . . . . . . . . . . . . . 35
4.4 The chordal graph sieve . . . . . . . . . . . . . . . . . . . . . . . 37
5 Recursive Schemes 44
5.1 Recursive schemes for semilattices . . . . . . . . . . . . . . . . . 44
5.2 Dynamic programming approach . . . . . . . . . . . . . . . . . . 45
VIII Contents
6 Reliability Applications 47
6.1 System reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.1.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.1.2 Abstract tubes for system reliability . . . . . . . . . . . . 48
6.1.3 Shier’s pseudopolynomial algorithm . . . . . . . . . . . . 49
6.1.4 Inclusion-exclusion and domination theory . . . . . . . . . 52
6.2 Special reliability systems . . . . . . . . . . . . . . . . . . . . . . 53
6.2.1 Communications networks . . . . . . . . . . . . . . . . . . 53
6.2.2 k-out-of-n systems . . . . . . . . . . . . . . . . . . . . . . 66
6.2.3 Consecutive k-out-of-n systems . . . . . . . . . . . . . . . 70
6.3 Reliability covering problems . . . . . . . . . . . . . . . . . . . . 75
6.3.1 The hypergraph model . . . . . . . . . . . . . . . . . . . . 75
6.3.2 Abstract tubes and polynomial algorithms . . . . . . . . . 75
6.4 Chapter notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7 Combinatorial Applications and Related Topics 82
7.1 Inclusion-exclusion on partition lattices . . . . . . . . . . . . . . 82
7.2 Chromatic polynomials and broken circuits . . . . . . . . . . . . 83
7.2.1 The usual chromatic polynomial . . . . . . . . . . . . . . 84
7.2.2 The generalized chromatic polynomial . . . . . . . . . . . 86
7.3 Sums over partially ordered sets. . . . . . . . . . . . . . . . . . . 89
7.3.1 A general theorem on sums . . . . . . . . . . . . . . . . . 89
7.3.2 Application to inclusion-exclusion . . . . . . . . . . . . . 90
7.4 Matroid polynomials and the β invariant . . . . . . . . . . . . . . 91
7.4.1 The Tutte polynomial . . . . . . . . . . . . . . . . . . . . 91
7.4.2 The characteristic polynomial . . . . . . . . . . . . . . . . 93
7.4.3 The β invariant . . . . . . . . . . . . . . . . . . . . . . . . 94
7.5 Euler characteristics and M¨obius functions . . . . . . . . . . . . . 95
7.5.1 Euler characteristics . . . . . . . . . . . . . . . . . . . . . 95
7.5.2 Mo¨bius functions . . . . . . . . . . . . . . . . . . . . . . . 96
Bibliography 100
Author Index 111
Subject Index 113
1 Introduction and Overview
Many problems in combinatorics, number theory, probability theory, reliabil-
ity theory and statistics can be solved by applying a unifying method, which
is known as the principle of inclusion-exclusion. The principle of inclusion-
exclusion expressesthe indicator function of a union of finitely many sets as an
alternating sum of indicator functions of their intersections. More precisely, for
anyfinite family ofsets{Av}v∈V the principleofinclusion-exclusionstatesthat
(cid:1) (cid:3) (cid:1) (cid:3)
(cid:2) (cid:4) (cid:5)
(1.1) χ Av = (−1)|I|−1χ Ai ,
v∈V I⊆V i∈I
I(cid:2)=∅
where χ(A) denotes the indicator function of A with respect to some universal
setΩ,thatis,χ(A)(ω)=1(cid:6)if(cid:7)ω ∈A,an(cid:8)dχ((cid:9)A)(ω)=0ifω(cid:6)∈(cid:7)Ω\A. E(cid:8)quivalently,
(1.1)canbe expressedasχ v∈V Av =(cid:7) I⊆V(−1)|I|χ i∈IAi ,where Av
denotes the complement of Av in(cid:0)Ω and i∈∅Ai = Ω. A proof by induc(cid:0)tion
on the number of sets is a common task in undergraduate courses. Usually,
the Av’s are measurable with respect to some finite measure µ on a σ-field of
subsets of Ω. Integration of the indicator function identity (1.1) with respect
to µ then gives
(cid:1) (cid:3) (cid:1) (cid:3)
(cid:2) (cid:4) (cid:5)
(1.2) µ Av = (−1)|I|−1µ Ai ,
v∈V I⊆V i∈I
I(cid:2)=∅
whichnowexpressesthemeasureofaunionoffinitelymanysetsasanalternat-
ing sum of measures of their intersections. The step leading from (1.1) to (1.2)
is referred to as the method of indicators [GS96b]. Naturally, two special cases
are of particular interest, namely the case where µ is the counting measure on
thepowersetofΩ,andthecasewhereµisaprobabilitymeasureonaσ-fieldof
subsets of Ω. These special cases are sometimes attributed to Sylvester (1883)
and Poincar´e (1896), although the second edition of Montmort’s book “Essai
d’Analyse sur les Jeux de Hazard”, which appeared in 1714, already contains
animplicituse ofthe method, basedona letter byN.Bernoulliin1710. A first
K.Dohmen:LNM1826,pp.1–4,2003.
(cid:1)c Springer-VerlagBerlinHeidelberg2003