The Lattice of all Betweenness Relations: Structure and Properties

Struture and properties

LaurentBeaudou,MamadouMoustaphaKanté,andLhouariNourine

ClermontUniversité,UniversitéBlaisePasal,LIMOS,CNRS,Frane

laurent.beaudouuniv-bplermont. fr, {mamadou.kante,nourine}isim a.fr

Abstrat. Weonsiderimpliationbaseswithpremisesofsizeexatly

2

^{, whih are alsoknownas} betweenness relations. Ourmotivations is thatseveralproblemsingraphtheoryanbemodelledusingbetweenness

relations,e.g.hullnumber,maximalliques.Inthispaperweharaterize

thelattieofallbetweennessrelationsbygivingitsposetofirreduible

elements.Moreover,weshowthatthislattieisameet-sublattieofthe

lattieofalllosuresystems.

1 Introdution

A onvexity spae on a ground set

X

^{is a subset of}

2 ^X

^{that is losed under}

intersetion. Convexity spaes were studied in [13℄ and are sometimes alled

Closuresystems.Themembersofaonvexityspaearealledonvexsets.Sine

the paper [13℄, onvexity spaes are studied by several authors who desribe

severaloftheirproperties(seethejointpaper[8℄ofEdelmanandJamisonfora

listofpubliationsduringtheeighties),inpartiularthesetofonvexityspaes

formsalattie.

Inthispaperwe dealwith betweenness relationswhihare speial asesof

onvexityspaes.Thenotionof betweennessrelationhasappearedintheearly

twentiethentury whenmathematiians fousedon fundamental geometry[4℄.

A betweenness relation

B

^{on a nite set}

X

^{is a set of triples}

(x, y, z ) ∈ X ³

^.

The mostintuitive betweennessrelationsare thoseomingfrommetri spaes

(apoint

y

^{is between}

x

^and

z

^{if theysatisfythetriangularequality).Aonvex}

set ofabetweennessrelation

B

^isasubset

Y

^of

X

^{suhthatforall}

(x, y, z ) ∈ B

^,

if

{x, z} ⊆ Y

^{, then}

y ∈ Y

^{. It is well-known that the set of onvex sets of}

a betweenness relation is a onvexity spae. Betweenness relations have been

thoroughlystudiedbyMengerandhisstudents[16℄.Otherbetweennessrelations

haveariseninresearheldsasprobabilitytheorywiththeworkofReihenbah

[18℄.Betweennessrelationshavebeenalsostudiedingraphsinordertogeneralize

geometrial theorems [1,2,9℄ (see the survey [17℄ for a non exhaustive list of

betweennessrelationsongraphs).

We are interested in desribing the set of all betweenness relations on a

groundset

X

^{.Weprovethatthesetof allonvexityspaeson}

X

^derivedfrom

betweennessrelationson

X

^{formsalattieandisinfata}meet-sublattieofthe lattie ofallonvexity spaeson

X

^{(wegiveanexample showingthatit isnot}

c 2012 by the paper authors. CLA 2012, pp. 317–325. Copying permitted only for private and academic purposes. Volume published and copyrighted by its editors.

Local Proceedings in ISBN 978–84–695–5252–0,

Universidad de M´ alaga (Dept. Matem´ atica Aplicada), Spain.

asub-lattie).Wealsodesribethe setofmeet andjoin-irreduibleelementsof

the lattie. We onlude by showing thatthe set of onvexity spaesobtained

fromlique betweenness relationsongraphs isa sublattie of the lattie ofall

onvexityspaesofbetweennessrelations.

Thispaperis motivatedby understandinglinksbetweenseveralparameters

that areonsidered in dierent areassuhasFCA,database, logi andgraph

theory.

Summary. NotationsanddenitionsaregiveninSetion2.Thedesriptionof

thelattieofonvexityspaesofbetweennessrelationsisgiveninSetion3.The

onvexityspaesofliquebetweennessrelationsisdesribedinSetion4.Some

questionsarisingfromalgorithmiaspetsaregiveninSetion5.

2 Preliminaries

Let

X

^{benite set.A partially ordered set on}

X

^{(orposet) isareexive,anti-}

symmetriandtransitivebinaryrelationdenotedby

P := (X, ≤)

^.For

x, y ∈ X

^,

we saythat

y

^overs

x

^,denotedby

x ≺ y

^,ifforany

z ∈ X

^with

x ≤ z ≤ y

^we

have

x = z

^or

y = z

^{.A lattie}

L := (X, ≤)

^{is apartially ordered setwith the}

followingproperties:

1. forall

x, y ∈ X

^{there existsaunique}

z

^{,denoted by}

x ∨ y

^{,suhthatforall}

t ∈ X

^,

t ≥ x

^and

t ≥ y

^implies

z ≤ t

^{.(Upperbound property.)}

2. forall

x, y ∈ X

^{there existsaunique}

z

^{,denoted by}

x ∧ y

^{,suhthatforall}

t ∈ X

^,

t ≤ x

^and

t ≤ y

^implies

z ≥ t

^{.(Lowerboundproperty.)}

Let

L = (X, ≤)

^{be a lattie. An element}

x ∈ X

^{is alled} join-irreduible (resp.meet-irreduible) if

x = y ∨ z

^(resp.

x = y ∧ z

^{) implies}

x = y

^or

x = z

^.

A join-irreduible(resp.meet-irreduible) element overs(resp. is overedby)

exatlyoneelement.Wedenoteby

J L

^and

M L

^thesetofalljoin-irreduibleand meet-irreduibleelementsof

L

respetively.

Theposetofirreduibleelementsofalattie

L = (X, ≤)

^isarepresentation of

L

^{byabipartiteposet}

Bip(L) = (J L , M L , ≤)

^{.Theoneptlattieof}

Bip(L)

is isomorphi to

L

^{(for moredetails see the books of DaveyandPriestley [3℄,}

andGanterandWille [10℄).

Animpliationon

X

^{isanordered pair}

(A, B)

^{of subsetsof}

X

^,denotedby

A → B

^{. The set}

A

^{is alled the premiseand the set}

B

^{the onlusion of the}

impliation

A → B

^.Let

Σ

^{be aset of} impliationson

X

^{.A subset}

Y ⊆ X

^is

Σ

^{-losedifforeahimpliation}

A → B

ⁱⁿ

Σ

^,

A ⊆ Y

^implies

B ⊆ Y

^.Thelosure

ofaset

S

^by

Σ

^,denotedby

S ^Σ

^{,isthesmallest}

Σ

^{-losedsetontaining}

S

^.

Let

S

^beasubsetof

X

^{.Algorithm1omputesthelosureof}

S

^byabetween-

ness relation

Σ

^{.It is knownas forward haining proedure or hase proedure}

[11℄.

Thesetof

Σ

^{-losedsubsetsof}

X

^,denotedby

F Σ

^{,isalosure system on}

X

Algorithm1:SetClosure(

S, Σ

⁾

Data:Aset

S ⊆ X

^and

Σ

^abetweenness Result:Thelosure

S ^Σ

begin

Let

S ^Σ := S

^;

while

∃xy → z ∈ Σ s.t. {x, y} ⊆ S ^Σ

^and

z / ∈ S ^Σ

^do

S ^Σ = S ^Σ ∪ {z}

^;

end

Conversely,givenalosuresystem

F

^on

X

^,afamily

Σ

^ofimpliationson

X

isalledanimpliationalbasisfor

F

^if

F = F Σ

^.Asubset

K ∈ X

^isalledakey

if

K ^Σ = X

^and

K

^{isminimalunderinlusionwiththisproperty.Thenamekey}

omesfromdatabasetheory[15℄.

Denition 1. Animpliationset

Σ

^on

X

^isalledabetweennessrelationiffor all

A → B ∈ Σ

^,

|A| = 2

^.

Twobetweennessrelations

Σ 1

^and

Σ 2

^aresaidtobeequivalent,denotedby

Σ ₁ ≡ Σ ₂

^,if

F _Σ 1 = F _Σ 2

^{.We denethe losure of a}betweennessrelation

Σ

^by

Σ ^c = {ab → c | a, b, c ∈ X

^and

Σ ≡ Σ ∪ {ab → c}}

^{. Note that}

Σ ^c

^{is the}

uniquemaximalbetweennessrelationequivalentto

Σ

^{.Ineahequivalenelass}

wedistinguishtwotypesofbetweennessrelations:

Canonial Aanonialbetweenness isthemaximuminitsequivalenelass.

Optimal Abetweenness

Σ

^{isoptimal ifforany}betweennessrelation

Σ ^′

^equiv-

alentto

Σ

^,wehave

|Σ| ≤ |Σ ^′ |

^.

A graph

G

^{is a pair}

(V (G), E(G))

^{, where}

V (G)

^{is the set of verties and}

E (G)

^{is theset ofedges.Weonsidersimplegraphs(forfurtherdenitionssee}

the book[6℄).Examplesofbetweennessrelationsarisingfromgraphtheoryare

Σ _G = {xy → z | z

^{liesinashortestpathfrom}

x

^toy

}

^and

Σ _G = {xy → V (G) | xy ∈ E(G)}

^{.Severalothernotionsofonvexityspaesaredenedongraphs(see}

[17℄). Figure1givesanexampleofagraph anditsonvexsetsforthe shortest

pathbetweenness.

3 The Lattie of all Betweenness Relations

Let

X

^{beanitesetand}

Σ

^abetweennessrelationon

X

^{.Giventwosets}

A

^and

C

ⁱⁿ

2 ^X

^suhthat

A ⊆ C

^{,wedenethe setinterval}

[A, C]

^{asthe familyofall}

sets

B

ⁱⁿ

2 ^X

^suhthat

A ⊆ B ⊆ C

^.

Demetrovisetal.[5℄gaveaharaterizationofonvexsetsofanimpliation

a b c

d

(a)Agraph

G

{}

{a,c}

{a}

{b,d}

{b,c}

{a,d}

{a,b,c,d}

{c} {d} {b}

(b) Convex sets of

G

^for

shortestpathbetweenness

Fig.1.Agraphanditsonvexsetsfortheshortestpathbetweennessrelation

Proposition1. [5 ℄Let

Σ

^bea betweennessrelation onaset

X

^.Then,

F Σ = 2 ^X \ [

ab→c∈Σ

[{a, b}, X \ {c}].

WedenotebyF

X := {F _Σ | Σ

^{is a}betweennessrelationon

X}

^thefamilyof

allbetweennessrelationson

X

^.

Theorem1. F

X

^{is a losure system and therefore a lattie when strutured}

underinlusion.

Proof. Wehavetoprovethatthisstrutureislosedundertheintersetionand

itontainsauniquemaximalelement.

Let

F 1 , F 2 ∈

^F

X

^{,thenthereexist}

Σ 1

^and

Σ 2

^{induingthesefamiliesofonvex}

setson

X

^.Let

F = F 1 ∩ F 2

^and

Σ

^bethebetweennessdenedby

ab → c ∈ Σ

^if

ab → c ∈ Σ 1

^or

ab → c ∈ Σ 2

^{.Thenwelaimthat}

F = F Σ

^.

Let

C

^beasetin

F

^{.Itisonvexfor}

Σ 1

^andfor

Σ 2

^{.Therefore,forany}

a, b

ⁱⁿ

C

^,every

c

^suhthat

ab → c ∈ Σ 1

^or

ab → c ∈ Σ 2

^isin

C

^{.Fromthis,wederive}

thatforany

a, b

ⁱⁿ

C

^,every

c

^suhthat

ab → c ∈ Σ

^isin

C

^,sothat

C

^isonvex

for

Σ

^.

Reiproally,let

C

^{beaonvexsetfor}

Σ

^{.Wewillshowthatitisonvexfor}

Σ 1

and

Σ 2

^.Let

a, b, c

^beelementsof

X

^suhthat

a

^and

b

^arein

C

^and

ab → c ∈ Σ 1

^.

Then

ab → c ∈ Σ

^andsine

C

^isonvexfor

Σ

^,

b

^{is in}

C

^.Therefore,

C

^{is in}

F 1

^.

Similarlyitisin

F 2

^{sothatitis in}

F

^.

Sine

Σ = Σ 1 ∪ Σ 2

^{,weonludethat}

Σ

^isabetweennessrelationandthere- forethesetF

X

^islosedunderintersetion.

Thefamily

2 ^X

^isinF

X

^{.Itariseswhen}

Σ

^istheemptybetweennessrelation.

Proposition2. Given two families

F 1

^and

F 2

^{in F}

X

^{and their anonial be-}

tweennessrelations

Σ ₁

^and

Σ ₂

^.Then

F ₁ ∧ F ₂ = F _Σ 1 ∪Σ 2

^.

F 1 ∨ F 2 = F Σ 1 ∩Σ 2

^.

Proof. See theproofofTheorem1fortherstpoint.Noww,notethat

Σ ₁

^and

Σ 2

^areanonialbetweennessrelations.Supposethat

F 1 ∨ F 2

^isnot

F Σ 1 ∩Σ 2

^.By

Proposition1,wehave

F 1 ∨ F 2 ⊆ F Σ 1 ∩Σ 2

^.

Nowall

Σ ∨

^{the anonial}betweennessrelationrelatedto

F 1 ∨ F 2

^.If

ab → c ∈ Σ ∨

^{for some}

a, b

^and

c

ⁱⁿ

X

^{, then we know that}

ab → c ∈ Σ i

^beause

F i ⊆ F 1 ∨ F 2

^for

i = 1, 2

^{(in the anonial form,we have every}

ab → c

^suh

that the orresponding interval has no intersetion with

F

^{). Therefore, every}

ab → c ∈ Σ ∨

^istruefor

Σ

^,and

F _Σ 1 ∩Σ 2 = F ₁ ∨ F ₂

^.

Corollary1. F

X

^isameet-sublattieofthelattieof alllosuresystemson

X

^.

Proof. Let

F 1

^and

F 2

^{betwofamiliesinF}

X

^.Sine

F 1 ∩ F 2

^{isthelosuresystem}

of abetweennessrelationthen themeetis preservedandthusF

X

^{is aameet-}

sublattie ofthe lattieofalllosuresystemson

X

^.

Remark 1. NotiethatF

X

^{isnotasublattieofthelattieofalllosuresystems}

on

X

^{.It sues toonsider the example where}

X = {1, 2, 3, 4}

^{.Takethe o-}

atoms

F 1 = 2 ^X \ [{1, 2}, {1, 2, 3}]

^denedbythe betweennessrelationrestrited to

12 → 4

^and

F 2 = 2 ^X \ [{2, 3}, {1, 2, 3}]

^denedbythe betweennessrelation restrited to

23 → 4

^.Then

F ₁ ∪ F ₂ = 2 ^X \ {{1, 2, 3}}

^{andis alosuresystem,}

while

F 1 ∨ F 2

^{isthetopelement,}

2 ^X

^.

Inthefollowing,wegiveaharaterizationoftheposetofirreduibleelements

ofF

X

^.

Proposition3. The poset of irreduible elements of F

X

^{is the bipartite poset}

Bip(

^F

X ) = (J

^F

X , M

^F

X , ⊆)

^where

J

F

X := {F ⊥ ∪ {S} | S ∈ 2 ^X \ F ⊥ }

^where

F ⊥ = {∅, X} ∪ {{x} | x ∈ X}

M

F

X := {2 ^X \ [ab, X \ {c}] | a, b, c ∈ X}.

Proof. Weprovethispropositionpoint bypoint.

Consider the maximal betweenness relation

Σ = {ab → c | a, b, c ∈ X}

^.

Then

F ⊥ = F Σ = 2 ^X \ S

ab→c∈Σ [{a, b}, X \ {c}]

^(see Proposition 1). Thus

F ⊥ = {∅, X} ∪ {{x} | x ∈ X}

^.

For meet-irreduible elements, we will rst onsider o-atoms. Let

Σ

^{be a}

betweennessrelation suh that

F Σ

^{is a o-atom. Sine}

Σ

^{is non-empty, then}

there exists a set whih is not onvex. Thus

Σ

^{must ontain an impliation}

ab → c

^,with

a, b, c ∈ X

^{,whihorrespondstothe maximallosuresystem in}

F

X

^{anddierent from}

2 ^X

^{.Namely,we removefrom}

2 ^X

^{the onvexsetsof the}

interval

[{a, b}, X \ {c}]

^.

Nowsupposethereexistsanothermeet-irreduibleelement

F

^whihisnota

o-atom. Call

Σ ₁

^a betweennessrelationsuh that

F

^is

F _Σ 1

^{. Also,all}

F ^′

^the

only suessorof

F _Σ 1

^and

Σ ₂

^abetweennesssuhthat

F ^′

^is

F _Σ 2

^{.By Proposi-}

tion 1,weknowthatfrom

F ^′

^to

F

^{,we removeatleastanintervalofthe form}

[{a, b}, X \ {c}]

^{.Thus, theo-atom assoiatedtothe impliation}

ab → c

^isnot

above

F ^′

^{butitis above}

F

^,sothat

F

^{has atleasttwosuessors.Weonlude}

thatallmeet-irreduibleelementsareo-atomsofF

X

^.

For join-irreduible elements, we will rst haraterizeatoms of F

X

^{. Pik}

any subset

S

^of

X

^{whihis not empty,a singletonor the whole of}

X

^{. Dene}

thebetweennessrelation

Σ _S

^suhthat

ab → c ∈ Σ _S

^forevery

a, b, c

ⁱⁿ

X

^exept

those where

a

^and

b

^{are in}

S

^and

c

^{is not in}

S

^.Then

F Σ S

^is

F ⊥ ∪ {S}

^.Now

supposethereexistsajoin-irreduible

F ∈

^F

X

^{thatis notanatom.}

F

^ontains

at least one set

S

^{whih is not in the unique losure system}

F ^′ ∈

^F

X

^that

it overs. But there is an atom whihontains exatly

F ” = F ⊥ ∪ {S}

^{, with}

F ” ⊆ F

^and

F ” 6⊆ F ^′

^.Thus

F

^{oversatleasttwoelements,andthus}

F

^isnota

join-irreduibleelement.

Corollary2. F

X

^ontains

n 2

(n − 2)2 ⁿ⁻³

meet-irreduible and

2 ⁿ − (n + 2)

join-irreduibleelements.

Proof. Everyo-atomisoftheform

2 ^X \[{a, b}, X \{c}]

^{andisaboveeveryatom}

formedby

F ⊥

^{andanysetnotinthe forbiddeninterval.Thismakes}

2 ⁿ − (n + 2) − 2 ⁿ⁻³

^{atomsbelowit.}

Anatomoftheform

F ⊥ ∪ S

^where

S

^isaseton

X

^ofsize2to

n − 1

^,isbelow

every o-atom thatdoes not forbid

S

^{. This number equals}

ⁿ ₂

(n − 2)2 ⁿ⁻³ − [ ^|S| ₂

(n − |S|)]

^.

Figure2showstheirreduibleposetfor

X = {1, 2, 3, 4}

^{whereeveryatomis}

represented bythe set

S

^addedto

F ⊥

^{andeveryo-atomis} representedby the removedimpliation

xy → z

^.

12 → 3 12 → 4 13 → 2 13 → 4 14 → 2 14 → 3 23 → 1 23 → 4 24 → 1 24 → 3 34 → 1 34 → 2

{1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} {1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4}

Fig.2.Irreduibleposetfor

n = 4

4 Clique Betweenness Relations

In this setion we deal with a speial betweenness relation dened througha

graph in a spei way. Many other betweenness relations on graphs an be

denedinthesameway,e.g.independentsets,setovers.

Givenagraph

G

^,wedenethebetweennessrelation

Σ G := {ab → c | c ∈ X, ab / ∈ E(G)}.

The onvexsetsof

Σ G

^{areexatlythe liquesof}

G

^{.Notie thatforanygraph}

G

^,thereisaorrespondingbetweennessrelation

Σ _G

^{.Inthe following,wehar-}

aterizethe lattieofall

Σ _G

^where

G

^{isa graphwithvertex-set}

X

^.Wedenote

byF

K X = {F Σ G | G

^isagraphon

X}

^.

Proposition4.

(

^F

^K _X , ⊆)

^isalattie.

Proof. Thebottom(resp.top)elementofF

X

^orrespondsto

F Σ G

^where

G

^isa

stable(resp.lique)on

X

^.

Moreover,F

K X

^islosedunderintersetion(theliquesof theintersetionof twographsareexatlythe onesinthe intersetionofboth familiesof liques).

Therefore,

(

^F

^K _X , ⊆)

^isalattie.

Sine

Σ G

^isabetweennessrelation,thenF

K X ⊂

^F

X

^{forany graph}

G

^dened

on

X

^.

Proposition5. Thelattie

(

^F

^K _X , ⊆)

^{isa sublattieof}

(

^F

X , ⊆)

^.

Proof. It is easy tosee thatitis ameet-sublattie,the meetof twofamiliesis

the intersetionofbothfamiliesinbothstrutures.

Inordertoprovethatitisajoin-sublattieof

(

^F

X , ⊆)

^{,onsidertwographs}

G 1

^and

G 2

^{andtheirliquefamilies}

F 1

^and

F 2

^.Call

G

^{thegraphunionof}

G 1

^and

G ₂

^and

F

^{thefamilyofitsliques.Therelated}betweennessrelationisobtained

bytakingthe intersetionof betweennessrelationsrelatedto

G ₁

^and

G ₂

^(non-

edgesin

G

^{areexatlynon-edgesin}

G ₁

^andin

G ₂

^{).Thereforeitisthejoinof}

F ₁

and

F 2

ⁱⁿ

(

^F

X , ⊆)

^.

Corollary3. Thelattie

(

^F

^K _X , ⊂)

^{isabooleanlattie with}

ⁿ ₂

atoms.

Proof. Foreahgraph

G

^,itsorrespondinganonialbetweennessrelationisthe set

Σ _G ^c := {ab → X | ab / ∈ E(G)}

^{.Notethatanysuper-setof}

Σ _G ^c

^orrespondsto

abetweennessrelationofapartialgraphof

G

^{,bydeletingedges}

ab

^from

G

^whih

orrespondstoadding

ab → X

ⁱⁿ

Σ _G ^c

^{.Sineanyatomof}

(

^F

^K _X , ⊂)

^orrespondsto

abetweennessrelationwhihontainsexatlyanimpliation

ab → X

^{,we have}

exatly

n 2

atoms.

5 AlgorithmiAspets of Betweenness Relations

In this setion, we reall some optimization problems related to betweenness

relations.

MinimumKey(MK)

Input:

Σ

^abetweennessrelationon

X

^and

k

^aninteger.

Question:Isthereaset

K ⊆ X

^suhthat

|K| ≤ k

^and

K ^Σ = X

^?

Theseproblems have been studiedin the severaldomains andspeially in

database theory,andthey have been provedNP-omplete [5,14,15℄for general

impliationbases.TheproblemMKhasbeenprovedNP-ompleteforpartiular

asesofbetweennessrelations(seeforinstane[7℄fortheshortestpathbetween-

nessrelationongraphs).Itisknownasthehullnumberofabetweennessrelation.

Therefore,wehavethefollowing.

Proposition6. MK isNP-omplete.

Reently,Kanté andNourine [12℄ have shown that

M K

^{is polynomial for}

shortest path betweennessrelationsof hordal anddistane hereditary graphs

byusingdatabasetehniques.CanweusethelattiestrutureofF

X

^togetnew

polynomialtimealgorithmsfortheMKprobleminnewgraphlasses?

Nowweonsidertheproblemwhihomputesanoptimaloverofabetween-

ness relation.Thisproblemis tond anoptimalbetweennessrelationwhihis

equivalenttoagivenbetweennessrelation.Severalworkshavebeendoneinthe

generalaseknownasHorn minimization[11℄.

Optimal Cover(OC)

Input:

Σ

^abetweennessrelationon

X

^and

k

^aninteger.

Question:Isthereabetweennessrelation

Σ ^′

^equivalentto

Σ

^suhthat

|Σ ^′ | ≤ k

^?

Thesizeofanoptimaloverisknownasthehydranumber [19℄.Theompu-

tationalomplexityof thehydranumberis openforbetweennessrelations,but

is NP-omplete forthegeneralase [11,15℄. Wehope thatthelattie struture

ofF

X

^{ouldhelptoaddressthisquestion.}

6 Conlusion

In thispaper, we haraterizethe ontextof the lattie of all betweennessre-

lations on a nite set, whih is a meet-sublattie of the lattie of all losure

systemsonthesameset.Weareonvinedthatthestrutureoflattieanhelp

tounderstand someproblemsof graph theorysuhashullnumber andhydra

number.Inthe future wewillinvestigatethe linkof theseparameters andthe

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