Enumeration of $2$-Polymatroids on up to Seven Elements

0.1 MB·

by  Thomas J. Savitsky

ENUMERATION OF 2-POLYMATROIDS ON UP TO SEVEN ELEMENTS
THOMASJ.SAVITSKY
ABSTRACT. Atheoryofsingle-elementextensionsofintegerpolymatroidsanalogoustothatofmatroidsisdevel-
4
oped.Wepresentanalgorithmtogenerateacatalogof2-polymatroids,uptoisomorphism.Whenweimplemented
1
thisalgorithmonacomputer,obtainingall2-polymatroidsonatmostsevenelements,wediscoveredthesurprising
0
factthatthenumberof2-polymatroidsonsevenelementsfailstobeunimodalinrank.
2

l
u
J
1. INTRODUCTION
9
1 A k-polymatroid is a generalization of a matroid in which the rank of an element may be greater than

1 but cannot exceed k. Precise definitions are given in the next section. Polymatroids have applications in
]
O mathematicsand computerscience. For example, Chapter11 of [6] employs2-polymatroidsin the studyof
C matchingtheory.Polymatroids,andmoregenerally,submodularfunctions,ariseincombinatorialoptimization;
seePartIVof[14]. Wetaketheperspectivethatk-polymatroidsareworthstudyingintheirownright.
.
h Although much work has been done with the use of computerson the enumerationof small matroids, to
t
a our knowledge, none has been done on enumeratingk-polymatroids, where k > 1. Some landmark results
m inmatroidenumerationincludethefollowing: in1973,Blackburn,Crapo,andHiggs[2]publishedacatalog
[ of all simple matroidson at most eightelements; in 2008, Mayhew and Royle [9] produceda catalog of all

matroidsonuptonineelements;andin2012,Matsumoto,Moriyama,Imai,andBremner[7]enumeratedall
2
rank-4matroidsontenelements.
v
6 Inthispaper,wedescribeoursuccessinadaptingtheapproachusedbyMayhewandRoyleto2-polymatroids.
0 Usingadesktopcomputer,weproducedacatalogofall2-polymatroids,uptoisomorphism,onatmostseven
0 elements.Weweresurprisedtodiscoverthatthenumberof2-polymatroidsonsevenelementsisnotunimodal
8
inrank.
.
1
0 2. BACKGROUND
4
For an introduction to polymatroids, we recommend Chapter 12 of [13]. We begin our discussion with
1
: definitions.
v
i Definition1. LetS beafiniteset. Supposeρ: 2S →Nsatisfiesthefollowingthreeconditions:
X
(i) ifX,Y ⊆S,thenρ(X ∩Y)+ρ(X ∪Y)≤ρ(X)+ρ(Y)(submodular),
r
a (ii) ifX ⊆Y ⊆S,thenρ(X)≤ρ(Y)(monotone),and
(iii) ρ(∅)=0(normalized).
Then(ρ,S)istermedanintegerpolymatroidorsimplyapolymatroidwithrankfunctionρandgroundsetS.
Definition2. Let k be a positive integer, andlet (ρ,S) be a polymatroid. Supposethatρ(x) ≤ k for every
x∈S. Then(ρ,S)isak-polymatroid.Amatroidmaybedefinedasa1-polymatroid.
Let(ρ,S)and(τ,T)bepolymatroids. Afunctionσ: S → T isanisomorphismofpolymatroidsifσ isa
bijectionandif ρ(X) = τ(σ(X)) foreveryX ⊆ S. Theclosure operatorof a polymatroidmaybe defined
exactlyasthatofamatroid.
Definition3. Theclosureoperatorcl:2S →2S ofapolymatroid(ρ,S)isgivenby
cl (X) = {x : ρ(X ∪x) = ρ(X)}forX ⊆ S. Thesetcl (X)iscalledtheclosureofX withrespecttoρ.
ρ ρ
Thesubscriptisomittedwhenρisclearfromcontext.
DEPARTMENTOFMATHEMATICS,THEGEORGEWASHINGTONUNIVERSITY,WASHINGTON,DC20052
E-mail address:[e-mail].
2 ENUMERATIONOF2-POLYMATROIDSONUPTOSEVENELEMENTS
Onecanshowthatρ(X)=ρ(cl(X))byinductionon|cl(X)−X|. Wewillfreelymakeuseofthisaswell
asthefollowingpropertiesofclosureoperators.Theyarestatedherewithoutproof.
Proposition4. Theclosureoperatorofapolymatroid(ρ,S)satisfiesthefollowingthreeproperties:
(i) X ⊆cl(X)forallX ⊆S (increasing),
(ii) ifX ⊆Y ⊆S,thencl(X)⊆cl(Y)(monotone),and
(iii) cl(X)=cl(cl(X))forallX ⊆S (idempotent).
Asubsetofthegroundsetthatismaximalwithrespecttorankiscalledaflat.Hereisthedefinitioninterms
oftheclosureoperator.
Definition5. Let(ρ,S)byapolymatroid. AsetX ⊆ S iscalledaflatofρifcl(X) = X. Thecollectionof
flatsof(ρ,S)issymbolizedbyF(ρ,S).
Intersectionsofflatsofmatroidsarethemselvesflats,andthesameistrueforpolymatroids.
Proposition6. IfF andGareflatsofpolymatroid(ρ,S),thenF ∩Gisalsoaflat.
Proof. Letx ∈S−(F ∩G). Eitherx ∈S−F orx ∈S−G. ByrelabelingF andGifnecessary,wemay
assumex∈S−F. Bysubmodularity,
ρ(F)+ρ((F ∩G)∪x)≥ρ(F ∪x)+ρ(F ∩G).
Thisimpliesρ((F ∩G)∪x)−ρ(F ∩G) ≥ ρ(F ∪x)−ρ(F). Byassumption,ρ(F ∪x)−ρ(F) > 0,and
hence,asneeded,ρ((F ∩G)∪x)−ρ(F ∩G)>0. (cid:3)
Since the entire groundset of a polymatroid is a flat, we see that the collection of flats of a polymatroid
formsalatticeunderset-inclusion.
Thetheoryofsingle-elementextensionsofmatroidswasdevelopedbyCrapoin[3]. Weextendthistheory
topolymatroidsinthenextsection,butfirstthematroidcaseisbrieflyreviewedhere. SeeSection7.2of[13]
foradetailedexposition.Webeginwithacoupleofdefinitionsthatapplytopolymatroidsaswell.
Definition7. Let(ρ,S)beapolymatroid,andletebeanelementnotinS. If(ρ¯,S∪e)isapolymatroidwith
ρ¯(X)=ρ(X)forallX ⊆S,thenρ¯isasingle-elementextensionofρ.
Definition8. Amodularcutofapolymatroid(ρ,S)isasubsetMofF(ρ,S)forwhich
(i) ifF ∈M,G∈F(ρ,S),andF ⊆G,thenG∈M,and
(ii) ifF,G∈Mandρ(F ∩G)+ρ(F ∪G)=ρ(F)+ρ(G),thenF ∩G∈M.
The nexttwo results show thatsingle-elementextensionsof a matroidcan be placedin one-to-onecorre-
spondencewithitsmodularcuts. Thiscorrespondenceunderliestheenumerationeffortsin[2]and[9].
Theorem9. Suppose(r,S)isamatroidwithsingle-elementextension(r¯,S∪e). Define
M={F ∈F(r,S):r(F)=r¯(F ∪e)}. ThenMisamodularcut.
Theorem10. Suppose(r,S)isamatroid,eisanelementnotinS,andM⊆F(r,S)isamodularcut.Define
r¯: 2S∪e →Nasfollows: forX ⊆S,setr¯(X)=r(X)and
r(X) if cl(X)∈M,
r¯(X ∪e)=
(r(X)+1 otherwise.
Then(r¯,S∪e)isamatroidandasingle-elementextensionof(r,S).
Ourfinaldefinitioninthissectionwillbeusedwhenwedescribetheflatsofsingle-elementextensions.
Definition11. LetF andGbeflatsofapolymatroid(ρ,S). SupposethatF (GandthatforanyflatH with
F ⊆H ⊆G,eitherH =F orH =G. ThenwesaythatGcoversF.
ENUMERATIONOF2-POLYMATROIDSONUPTOSEVENELEMENTS 3
3. SINGLE-ELEMENTEXTENSIONSOF POLYMATROIDS
Givenapolymatroid,ouraimistodescribeallofitssingle-elementextensions. Asinthematroidcasewe
mayrestrictourattentiontoflatsoftheoriginalpolymatroid.Suppose(ρ¯,S∪e)isasingle-elementextension
of(ρ,S). Thefollowingpropositionshowsthatifthevalueofρ¯(F ∪e)isknownforeveryflatF of(ρ,S),
thenρ¯iscompletelydetermined.
Proposition12. Suppose(ρ¯,S∪e)isasingle-elementextensionof(ρ,S). LetX ⊆ S,andletcl(X)bethe
closureofX withrespecttoρ(notρ¯). Thenρ¯(X ∪e)=ρ¯(cl(X)∪e).
Proof. SinceX ∪e⊆cl(X)∪e=clρ¯(X)∪e⊆clρ¯(X ∪e)andρ¯hasthesamevalueonthefirstandlastof
thesesets,theresultfollows. (cid:3)
Forasingle-elementextension(ρ¯,S∪e)of(ρ,S),letcbeρ¯(e)andletX ⊆S. Itfollowsthatρ¯(X∪e)≤
ρ(X)+c by the submodularityand normalizationof ρ¯. Therefore, we may partition the flats of (ρ,S) into
classesM0,M1,...,McbytheruleF ∈Miifandonlyifρ¯(F∪e)=ρ(F)+i.(NotethatsomeMimaybe
empty.)ByProposition12,knowledgeof(ρ,S)andthepartition(M0,M1,...,Mc)completelydetermines
(ρ¯,S∪e). Ourgoalistodeveloppropertiesthatcharacterizesuchpartitions. Thefollowingdefinitionwillbe
useful.
Definition13. Let(ρ,S)beapolymatroid,andletX,Y ⊆S. DefinethemodulardefectofX andY,denoted
δ(X,Y),tobeρ(X)+ρ(Y)−ρ(X ∪Y)−ρ(X ∩Y). Ifδ(X,Y)=0,thenX andY areamodularpairof
sets.
Now suppose (M0,M1,...,Mc) is a partition of F(ρ,S). Let e be an element not in S and define
ρ¯: 2S∪e →Nasfollows: forX ⊆ S,setρ¯(X)= ρ(X)and,ifcl(X)∈ M ,thensetρ¯(X ∪e)= ρ(X)+i.
i
Furthermore,defineafunctionµ: 2S →Nbyµ(X)=iifcl(X)∈M .
i
Theorem14. Asdefinedabove,(ρ¯,S∪e)isapolymatroid,andhenceasingle-elementextensionof(ρ,S),if
andonlyifthefollowingthreeconditionsholdforallflatsF,Gof(ρ,S):
(I) µ(F ∩G)+µ(F ∪G)−δ(F,G)≤µ(F)+µ(G),
(II) ifF ⊆G,thenρ(F)+µ(F)≤ρ(G)+µ(G),and
(III) ifF ⊆G,thenµ(G)≤µ(F).
Proof. Assume(ρ¯,S∪e)isapolymatroid,andletF,Gbeflatsof(ρ,S). Applyingthesubmodularityofρ¯to
thepairofsetsF ∪eandG∪egives
ρ¯((F ∪e)∩(G∪e))+ρ¯((F ∪e)∪(G∪e))≤ρ¯(F ∪e)+ρ¯(G∪e).
Byourdefinitionofρ¯,therightsideoftheaboveinequalityequalsρ(F)+µ(F)+ρ(G)+µ(G).Theleftside
equals
ρ¯((F ∩G)∪e)+ρ¯((F ∪G)∪e)=ρ(F ∩G)+µ(F ∩G)+ρ(F ∪G)+µ(F ∪G)
=µ(F ∩G)+µ(F ∪G)+ρ(F)+ρ(G)−δ(F,G).
Weconcludethatµ(F ∩G)+µ(F ∪G)−δ(F,G)≤µ(F)+µ(G)andseethatcondition(I)issatisfied.
Statement(II)isthemonotonepropertyofρ¯.
Finally,toshowcondition(III),applythesubmodularityofρ¯tothepairofsetsF ∪eandG. Thisgivesthe
firstofthefollowingequivalentinequalities:
(1) ρ¯((F ∪e)∪G)+ρ¯((F ∪e)∩G)≤ρ¯(F ∪e)+ρ¯(G)
(2) ρ¯(G∪e)+ρ¯(F)≤ρ¯(F ∪e)+ρ¯(G)
(3) ρ¯(G∪e)−ρ¯(G)≤ρ¯(F ∪e)−ρ¯(F)
(4) µ(G)≤µ(F).
Nowassumethatconditions(I),(II),and(III)aresatisfied. Wemustverifythatρ¯satisfiesthethreeaxioms
forapolymatroid.Itfollowsimmediatelyfromourdefinitionthatρ¯(∅)=0.
Next, we checkmonotonicity. Assume thatX ⊆ Y ⊆ S. The definitionof ρ¯and the monotonicityof ρ
implythatρ¯(X) = ρ(X) ≤ ρ(Y) = ρ¯(Y). Thuswealsogetρ¯(X) ≤ ρ¯(Y) ≤ ρ¯(Y ∪e). Itremainstocheck
4 ENUMERATIONOF2-POLYMATROIDSONUPTOSEVENELEMENTS
thatρ¯(X ∪e)≤ρ¯(Y ∪e). Observe
ρ¯(X ∪e)=ρ(X)+µ(X)
=ρ(cl(X))+µ(cl(X))
≤ρ(cl(Y))+µ(cl(Y)) (bycondition(II))
=ρ(Y)+µ(Y)
=ρ¯(Y ∪e).
Therefore,ρ¯ismonotoneonallsubsetsofS∪e.
Sinceρ¯(X) = ρ(X)forX ⊆ S,tochecksubmodularityitsufficestoverifyitforthepairs(a)X ∪eand
Y,and(b)X ∪eandY ∪e,withX,Y ⊆S. Forcase(a),wehave
ρ¯((X ∪e)∩Y)+ρ¯((X ∪e)∪Y)=ρ¯(X ∩Y)+ρ¯((X ∪Y)∪e)
=ρ(X ∩Y)+ρ(X ∪Y)+µ(cl(X ∪Y))
≤ρ(X)+ρ(Y)+µ(cl(X ∪Y)) (bythesubmodularityofρ)
≤ρ(X)+ρ(Y)+µ(cl(X)) (bycondition(III))
=ρ¯(X ∪e)+ρ¯(Y).
Forcase(b),wehave
ρ¯(X ∪e)+ρ¯(Y ∪e)=ρ(cl(X))+µ(cl(X))+ρ(cl(Y))+µ(cl(Y))
≥µ(cl(X)∩cl(Y))+µ(cl(X)∪cl(Y))−δ(cl(X),cl(Y))+ρ(cl(X))+ρ(cl(Y))
=µ(cl(X)∩cl(Y))+µ(cl(X)∪cl(Y))+ρ(cl(X)∪cl(Y))+ρ(cl(X)∩cl(Y))
=ρ¯((cl(X)∪cl(Y))∪e)+ρ¯((cl(X)∩cl(Y))∪e)
≥ρ¯(X ∪Y ∪e)+ρ¯((X ∩Y)∪e).
Thefirst inequalityfollowsbycondition(I),and the last inequalityholdsbecause the monotonicityof ρ¯has
alreadybeenestablished. (cid:3)
NotethatTheorem14generalizesTheorems9and10forsingle-elementextensionsofmatroids. Alsonote
that if the conditions of the theorem are satisfied, then M0 is a modular cut. Lastly, we point out that the
theoremremainstrueiftheword“flats”isreplacedby“sets”initsstatement.
Definition15. Apartition(M0,M1,...,Mc)offlatsofapolymatroid(ρ,S)thatsatisfiestheconditionsin
Theorem14iscalledanextensiblepartition.
Fortheremainderofthissection,assumethat(ρ,S)isapolymatroid,(M0,M1,...,Mc)isanextensible
partition,and(ρ¯,S ∪e)isthesingle-elementextensiondefinedrightbeforeTheorem14. Ournextgoalisto
describetheflatsof(ρ¯,S∪e).
ClearlyifF ⊆ S isaflatof(ρ¯,S ∪e),thenF isalsoaflatof(ρ,S). We alsohavethefollowinghelpful
fact.
Proposition16. ForF ⊆S,ifF ∪eisaflatof(ρ¯,S∪e),thenF isaflatof(ρ,S).
Proof. Observethatclρ(F)⊆clρ¯(F)⊆clρ¯(F ∪e)=F ∪e. (cid:3)
Therefore,tofindtheflatsof(ρ¯,S∪e)weneedonlyconsidersetsoftheformF andF ∪e,whereF isa
flatof(ρ,S). Thenextpropositionexplicitlydescribestheflatsof(ρ¯,S∪e).
Proposition17. Let(ρ¯,S∪e)bethesingle-elementextensionof(ρ,S)correspondingtotheextensibleparti-
tion(M0,M1,...,Mc). Theflatsof(ρ¯,S∪e)arethesets
(1) F inM ,fori>0,
i
(2) F ∪e,forF ∈M0,
(3) F ∪e,forF ∈M withi>0,whereF hasnocoverGwithρ(F)+µ(F)=ρ(G)+µ(G).
i
ENUMERATIONOF2-POLYMATROIDSONUPTOSEVENELEMENTS 5
Proof. Toreiterate,weneedonlylookatsetsoftheformF andF ∪e,whereF isaflatof(ρ,S).
Itfollowsfromthedefinitionofρ¯thataflatF of(ρ,S)isaflatof(ρ¯,S∪e)ifandonlyifF 6∈M0.
IfF ∈M0,thenF ∪eisaflatof(ρ¯,S∪e)since,fory ∈S−F,wehave
ρ¯(F ∪{e,y})≥ρ(F ∪y)>ρ(F)=ρ¯(F ∪e).
We claim that for F ∈ M with i > 0, the set F ∪e is a flat of (ρ¯,S ∪e) if and only if the inequality
i
in property (II) of Theorem 14 is strict for all covers G of F. Indeed, if G covers F and ρ(F)+µ(F) =
ρ(G)+µ(G),thenF ∪eisnotaflatsince
ρ¯(F ∪e)=ρ(F)+µ(F)=ρ(G)+µ(G)=ρ¯(G∪e).
Nowassumestrictinequalityholdsinproperty(II)forallcoversofF. Ifx ∈ S−F,thenthereisacoverG
ofF withF (G⊆cl (F ∪x),so
ρ
ρ¯(F ∪e)=ρ(F)+µ(F)<ρ(G)+µ(G)=ρ¯(G∪e)≤ρ¯(F ∪{e,x}). (cid:3)
Notethatifµ(G)=µ(F),thenequalitycannotholdinproperty(II)ofTheorem14,sinceρ(G)>ρ(F).
Theseresultsgeneralizethoseformatroidextension. We definethecollarofM toconsistofeveryF ∈
i
M that is covered by some G ∈ M with j < i. In a matroid (r,S), if a flat G covers a flat F, then
i j
r(G)−r(F)=1. If(r¯,S∪e)isasingle-elementextensionandF ∈M1,thenF ∪eisaflatofr¯ifandonly
ifF isnotinthecollarofM1.
4. GENERATINGA CATALOGOF SMALL2-POLYMATROIDS
Nowwewillspecializetheresultsoftheprevioussectionto2-polymatroids.
Suppose(ρ,S) isa 2-polymatroidwithcollectionofflatsF(S). SupposethatF(S) istheunionof three
disjointsets, M0, M1, andM2, some of which may be empty. Let e be an elementnotin S. We define a
functionρ¯: 2S∪e →Nasfollows. ForX ⊆S,defineρ¯(X)=ρ(X)and
ρ¯(X ∪e)=ρ(X)+i where cl(X)∈M .
i
When computing the extensible partitions of a 2-polymatroid, we found it convenient to work with the
followingverbosespecializationofTheorem14.
Theorem18. Asdefined,(ρ¯,S ∪e)is a2-polymatroidextensionof(ρ,S)ifandonlyifthe followingseven
conditionsaremet.
(1) IfF ∈M2,G∈F(S),F ⊆G,andρ(G)−ρ(F)=1,thenG∈M1∪M2. Inotherwords,ifF ∈M2
iscoveredbyaflatGof(ρ,S)ofonerankhigher,thenGcannotbeinM0.
(2) IfF,G∈M0and(F,G)isamodularpair,thenF ∩G∈M0aswell.
(3) IfF,G∈M0andρ(F)+ρ(G)=ρ(F ∪G)+ρ(F ∩G)+1,thenF ∩G∈M0∪M1.
(4) If F,G ∈ M1 and (F,G) is a modular pair, then either F ∩G ∈ M1 as well, or F ∩G ∈ M2 and
cl(F ∪G)∈M0.
(5) IfF ∈M0,G∈M1,and(F,G)isamodularpair,thenF ∩GcannotbeinM2.
(6) ThesetM2isdown-closedinthelatticeF(ρ,S).
(7) ThesetM0isup-closedinthelatticeF(ρ,S).
SketchofProof. Condition(II)ofTheorem14specializestocondition(1)here,condition(I)toconditions(2)
through(5),andcondition(III)toconditions(6)and(7). (cid:3)
Theflatsof(ρ¯,S∪e)arethesets
(1) F inM1∪M2,
(2) F ∪e,forF ∈M0,
(3) F ∪e,forF ∈M ,withi>0,where,
i
(a) F hasnocoverGinMi−1withρ(G)=ρ(F)+1,and
(b) ifi=2,F hasnocoverGinM0withρ(G)=ρ(F)+2.
6 ENUMERATIONOF2-POLYMATROIDSONUPTOSEVENELEMENTS
Forexample,let(ρ,{a,b})bethe2-polymatroidconsistingoftwolinesplacedfreelyinaplane.Tobespe-
cific,defineρ(∅)=0,ρ({a})=ρ({b})=2,andρ({a,b})=3. Thesingle-elementextensioncorresponding
totheextensiblepartition
(M0,M1,M2)=({{a,b}},{{a},{b}},{∅})
isthe2-polymatroidconsistingofthreelinesplacedfreelyinaplane.
Usingtheresultsofthissection,weendeavoredtocatalogallsmall2-polymatroidsonacomputerbymeans
ofacanonicaldeletionalgorithm.
Definition19. SupposeX isacollectionofcombinatorialobjectswithgroundset{1,...,n}andanotionof
isomorphism.AfunctionC: X →X isacanonicallabelingfunctionifthefollowingholdforallX,Y ∈X:
(i) X isisomorphictoC(X),and
(ii) C(X)=C(Y)ifandonlyifX isisomorphictoY.
Inthiscase,C(X)iscalledthecanonicalrepresentativeofX.
BrendanMcKay’snautyprogramefficientlycomputescanonicallylabelingsofcoloredgraphs. Inorder
tomakeuseofit,weconvertpolymatroidsintographsusingthefollowingconstruction.
Definition20. Givenanintegerpolymatroid(ρ,S), defineacolored,bipartitegraphwith bipartitionS and
F(ρ,S). Anedgebetweene ∈ S andF ∈ F(ρ,S)exists ifandonlyife ∈ F. ColorF ∈ F(ρ,S)withits
rank,ρ(F). Coloreache∈S with−1. Calltheresultinggraphtheflatgraph1oftheintegerpolymatroid.
NotethatifX ⊆SandifF isthesmallestflatcontainingX,thenρ(X)=ρ(F). Intermsoftheflatgraph,
therankofasetX ⊆ S equalstheleastcoloramongstthoseverticesadjacenttoeveryelementofX. Using
thisobservation,itiseasytoprovethenextproposition.
Proposition21. Two integer polymatroidsare isomorphic if and only if their flat graphs are isomorphic as
coloredgraphs.(Byanisomorphismofacoloredgraph,wemeanagraphisomorphismthatmapseachvertex
toanotherofthesamecolor.)
Therefore,inordertocanonicallylabela2-polymatroid,itsufficestoconsideritsflatgraph. Thennauty
isusedtocomputeacanonicallabelingoftheflatgraph.Whenrestrictedtothegroundsetofthepolymatroid,
thisgivesacanonicallabelingofthepolymatroid.Foradescriptionofthealgorithmsusedbynautysee[10]
and[11]. Onemayalsofindtheexpositionin[5]helpful.
NowwehaveallthetoolsneededtoadaptAlgorithm1of[9]to2-polymatroids. Supposewearegivena
setX thatconsistsofpreciselyonerepresentativeofeachisomorphismclassof2-polymatroidsontheground
n
set{1,...,n}. Thefollowingalgorithmproducesitscounterpart,Xn+1,fortheset{1,...,n+1}.
Algorithm1Isomorph-freegenerationof2-polymatroids
foreachρ∈X do
n
SetYρ ←∅,thecollectionofextensionsofρthatshouldappearinXn+1.
foreachextensiblepartition(M0,M1,M2)ofρdo
Letρ¯betheextensionofρassociatedwiththispartition.
Canonicallylabelρ¯.
Setρ′ ←ρ¯\(n+1),thecanonicaldeletion.
Canonicallylabelρ′.
ifρ=ρ′andρ¯6∈Y then
ρ
SetY ←Y ∪ρ¯.
ρ ρ
endif
endfor
SetXn+1 ←Xn+1∪Yρ.
endfor
return X
n+1
1Inourimplementation,wefounditprudenttoinsertanisolatedvertexofrankrifnoflatsofrankrexisted,forr<ρ(S).Thismade
iteasiertoworkwiththelabelingsusedbynauty.
ENUMERATIONOF2-POLYMATROIDSONUPTOSEVENELEMENTS 7
Afewcommentsareinorder. Notethatthetestρ = ρ′ isforequality,notisomorphism. Inourimplemen-
tation,thecollectionsY arebinarytrees,ratherthanmerelysets,inordertospeedupthesearchρ∈Y .
ρ ρ
The task of finding all extensible partitions for a polymatroidρ is relatively straightforward, but tedious.
First,acandidateforamodularcutM0isfound.SinceM0isanup-closedset,itsufficestokeeptrackofthe
minimalflatsinM0. Thesearefoundasindependentsetsofagraphwithvertexsetequaltotheflatsofρ. If
oneflatiscontainedinanother,anedgeisplacedbetweenthetwo. Condition(2)ofTheorem18isthenused
tonarrowthesearch. Anedgeisalsoplacedbetweenanytwoflatsthatformamodularpair. Theindependent
setsinthisgrapharetheminimalmembersofourcandidatesforM0.GivenanacceptablecandidateforM0,a
morecomplicatedprocedureisusedtofindallpossiblecandidatesforM1. Theremainingflatsareobviously
assigned to M2. Unfortunately, the resulting partition must be checked to see if it satisfies conditions (1)
through(5),sincesomeofthesemayfailfornon-minimalmembersofM0orM1.
Finally, note that each iteration of the outermostfor loop may be run in parallel since extensions of two
differentmembersofX areneverdirectlycomparedtoeachother.
n
5. IMPLEMENTATIONAND RESULTS
WeimplementedthisalgorithmintheCprogramminglanguage.Inordertodeterminethecoverrelationsfor
flats,weemployedtheATLASlibrary[16]tomultiplytheadjacencymatricesofgraphs.Weusedtheigraph
library [4] to find independent sets in graphs. A computer with a single 6-core Intel i7-3930K processor
clockedat 3.20GHzrunning64-bitUbuntuLinuxexecutedthe resulting program. After approximatelyfour
days,acatalogofall2-polymatroidsonsevenorfewerelementswasgenerated.
Thefollowingtableliststhenumberof2-polymatroids,uptoisomorphism,onthegroundset{1,...,n},
byrank.
Thenumberofunlabeled2-polymatroids
rank\n 0 1 2 3 4 5 6 7
0 1 1 1 1 1 1 1 1
1 1 2 3 4 5 6 7
2 1 4 10 21 39 68 112
3 2 12 49 172 573 1890
4 1 10 78 584 5236 72205
5 3 49 778 18033 971573
6 1 21 584 46661 149636721
7 4 172 18033 19498369
8 1 39 5236 149636721
9 5 573 971573
10 1 68 72205
11 6 1890
12 1 112
13 7
14 1
total 1 3 10 40 228 2380 94495 320863387
Thefollowingpropositionisthekeytoproducingtheanalogoustableforlabeled2-polymatroids.
Proposition22. Theautomorphismsofanintegerpolymatroid(ρ,S)areinone-to-onecorrespondencewith
theautomorphismsofitsflatgraph.
SketchofProof. Thisisnothardtoshow.Itfollows,forexample,fromtheremarksinSection1of[12],which
employsthelanguageofhypergraphs. (cid:3)
Sincenautycaneasilycomputetheautomorphismgroupsoftheflatgraphsofthesepolymatroids,apply-
ingthe Orbit-StabilizerTheoremgivesa countof the numberoflabeled 2-polymatroidson 7 elements. The
followingtableliststhenumberoflabeled2-polymatroids,onthegroundset{1,...,n},byrank.
8 ENUMERATIONOF2-POLYMATROIDSONUPTOSEVENELEMENTS
Thenumberoflabeled2-polymatroids
rank\n 0 1 2 3 4 5 6 7
0 1 1 1 1 1 1 1 1
1 1 3 7 15 31 63 127
2 1 6 29 135 642 3199 16879
3 3 41 477 5957 87477 1604768
4 1 29 784 27375 1554077 189213842
5 7 477 41695 7109189 3559635761
6 1 135 27375 21937982 733133160992
7 15 5957 7109189 86322358307
8 1 642 1554077 733133160992
9 31 87477 3559635761
10 1 3199 189213842
11 63 1604768
12 1 16879
13 127
14 1
total 1 3 14 115 2040 109707 39445994 1560089623047
The symmetry of the columns in the above tables is explained by the following notion of duality for k-
polymatroids.
Definition23. Givenapolymatroid(ρ,S),definethek-dualρ∗: 2S →Nby
∗
ρ (X)=k|X|+ρ(S−X)−ρ(S).
Itiseasilyseenthatρ∗ isitselfak-polymatroidandthattheoperationofk-dualityisaninvolutiononthe
setofk-polymatroidsonafixedgroundsetwhichrespectsisomorphism. (Infact,itisshownin[17]tobethe
theuniquesuchinvolutionthatinterchangesdeletionandcontraction.)
Welshconjecturedthatthenumberofmatroidsonafixedsetisunimodalinrankin[15]. Thecounterpartof
thisconjecturefork-polymatroidsisfalse. Thetableaboveshowsthatitfailsfor2-polymatroidson7elements.
Sincethenumberoflabeled2-polymatroidsonsevenelementsisnearlyafactorof7!morethanthenum-
ber of unlabeled ones, it seems reasonable to conjecture that, asymptotically, almost all 2-polymatroidsare
asymmetric.
Theproofin [8]thatalmostallmatroidsarelooplesscarriesoverwithoutchangeto 2-polymatroids. Our
catalogsuggeststhatastrongerpropertyholdsfor2-polymatroids. Weconjecturethat,asymptotically,almost
all2-polymatroidscontainnoelementsofranklessthan2. Hereistheevidencefromourcatalog:thenumber
ofunlabeled2-polymatroidson{1,...,n}withnoelementsofranklessthan2.
n 1 2 3 4 5 6 7
count 1 2 8 51 696 49121 304541846
Thistableshouldbecomparedtothefirsttableinthissection.
6. A CONFIRMATION
Considerthatthelabeledsingle-elementextensionsofak-polymatroidareinfactsolutionstoacertaininte-
gerprogrammingproblem.Whenallsubsetsofthegroundsetaretakenasvariables,inequalitiesguaranteeing
theaxiomsofak-polymatroidareeasilywritten. Tobeconcrete,letρ: S → Nbeak-polymatroidandlete
beanelementnotinS. Regardρ¯(X)asavariableforeachX ⊆ S ∪e. Fixρ¯(A) = ρ(A)forA ⊆ S. Also
fixρ¯(S∪e)=ρ(S)+c,wherecisanaturalnumbernogreaterthank. Nownonnegativeintegersolutionsto
thesystemofinequalitiesbelowareinone-to-onecorrespondencewithlabeledsingle-elementextensionsofρ
whichincreasetherankofρbyc.
ENUMERATIONOF2-POLYMATROIDSONUPTOSEVENELEMENTS 9
ρ¯(A)+ρ¯(A∪f ∪g)≤ρ¯(A∪f)+ρ¯(A∪g) forA⊆S∪eandf,g ∈(S∪e)−A;
0≤ρ¯(A∪f)−ρ¯(A)≤k forA⊆S∪eandf ∈(S∪e)−A; and
ρ¯(A)≤k|A| forA⊆S∪e.
Here,weareusingaconditionequivalenttosubmodularity;seeTheorem44.2of[14]foraproofofequiv-
alence. The open-sourceoptimization software SCIP [1] is able to countthe number of integer solutions to
suchinequalities. UsingSCIPwe verifiedthenumbersoflabeled2-polymatroidsgivenearlier. Notethat, in
theversionofSCIPweusedinApril2013,itwasnecessarytoturnoffallpre-solvingoptionsinordertoob-
tainaccurateresults. Thisprocesstookapproximately13weeksusingthecomputerdescribedintheprevious
section.
7. ACKNOWLEDGEMENTS
Theauthorwouldliketothanktheanonymousrefereeforcarefullyreadingthispaperandprovidingsug-
gestionswhichimproveditsexposition.
REFERENCES
[1] TobiasAchterberg.Scip:Solvingconstraintintegerprograms.MathematicalProgrammingComputation,1(1):1–41,July2009.
[2] JohnE.Blackburn, HenryH.Crapo, andDenis A.Higgs. Acatalogue ofcombinatorial geometries. Math. Comp., 27:155–166;
addendum,ibid.27(1973),no.121,loosemicrofichesuppl.A12–G12,1973.
[3] HenryH.Crapo.Single-elementextensionsofmatroids.J.Res.Nat.Bur.StandardsSect.B,69B:55–65,1965.
[4] Ga´borCsa´rdiandTama´sNepusz.Theigraphsoftwarepackageforcomplexnetworkresearch.InterJournalComplexSystems,page
1695,2006.
[5] StephenG.HartkeandA.J.Radcliffe.McKay’scanonicalgraphlabelingalgorithm.InCommunicatingMathematics,volume479
ofContemp.Math.,pages99–111.Amer.Math.Soc.,Providence,RI,2009.
[6] La´szlo´Lova´szandMichaelD.Plummer.MatchingTheory.AMSChelseaPublishing,Providence,RI,2009.Correctedreprintofthe
1986original.
[7] YoshitakeMatsumoto,SonokoMoriyama,HiroshiImai,andDavidBremner.Matroidenumerationforincidencegeometry.Discrete
Comput.Geom.,47(1):17–43,2012.
[8] DillonMayhew,MikeNewman,DominicWelsh,andGeoffWhittle.Ontheasymptoticproportionofconnectedmatroids.European
J.Combin.,32(6):882–890,2011.
[9] DillonMayhewandGordonF.Royle.Matroidswithnineelements.J.Combin.TheorySer.B,98(2):415–431,2008.
[10] BrendanD.McKay.Practicalgraphisomorphism.InProceedingsoftheTenthManitobaConferenceonNumericalMathematicsand
Computing,Vol.I(Winnipeg,Man.,1980),volume30,pages45–87,1981.
[11] BrendanD.McKayandAdolfoPiperno.Practicalgraphisomorphism,II.J.SymbolicComput.,60:94–112,2014.
[12] GaryL.Miller.Isomorphismofgraphswhicharepairwisek-separable.Inform.andControl,56(1-2):21–33,1983.
[13] JamesOxley.MatroidTheory,volume21ofOxfordGraduateTextsinMathematics.OxfordUniversityPress,Oxford,secondedition,
2011.
[14] AlexanderSchrijver.Combinatorial optimization.Polyhedraandefficiency. Vol.B,volume24ofAlgorithmsandCombinatorics.
Springer-Verlag,Berlin,2003.Matroids,trees,stablesets,Chapters39–69.
[15] D.J.A.Welsh.Combinatorialproblemsinmatroidtheory.InCombinatorialMathematicsanditsApplications(Proc.Conf.,Oxford,
1969),pages291–306.AcademicPress,London,1971.
[16] R.ClintWhaleyandAntoinePetitet.MinimizingdevelopmentandmaintenancecostsinsupportingpersistentlyoptimizedBLAS.
Software:PracticeandExperience,35(2):101–121,February2005.
[17] GeoffWhittle.Dualityinpolymatroidsandsetfunctions.Combin.Probab.Comput.,1(3):275–280,1992.