Forbidden Configurations: Finding the number predicted by the Anstee-Sali Conjecture is NP-hard

0.28 MB·English

by  Miguel Raggi

DiscreteMathematicsandTheoreticalComputerScience DMTCSvol. (subm.),bytheauthors,1–1
Forbidden Configurations: Finding the
number predicted by the Anstee-Sali
Conjecture is NP-hard
2
1
0 Miguel Raggi †
2

t CentrodeCienciasMatema´ticas
c
UniversidadNacionalAuto´nomadeMe´xico,
O
Morelia,Michoaca´n,Me´xico

0
3

] LetF beahypergraphandletforb(m,F)denotethemaximumnumberofedgesahypergraphwithmverticescan
M
haveifitdoesn’tcontainF asasubhypergraph. AconjectureofAnsteeandSalipredictstheasymptoticbehaviour
D offorb(m,F)forfixedF. InthispaperweprovethatevenfindingthispredictedasymptoticbehaviourisanNP-
hardproblem,meaningthatiftheAnstee-Saliconjectureweretrue,findingtheasymptoticsofforb(m,F)wouldbe
.
s NP-hard.
c
[
MathematicsSubjectClassification:05D05

1
v Keywords:ForbiddenConfiguration,Hypergraph,Trace,NP-hard,NP-complete,Anstee-Saliconjecture
9
8
1
8 1 Introduction
.
0
Thepaperconsidersanextremalproblem.SomeofthemostcelebratedextremalresultsarethoseofErdo˝s
1
andStone[ES46]andErdo˝sandSimonovits[ES66].Theyconsiderthefollowingproblem:Givenm∈N
2
1 andagraphF,findthemaximumnumberofedgesinagraphonmverticesthatavoidshavingasubgraph
: isomorphictoF.
v
i Thereareanumberofwaystogeneralizethistohypergraphs. Ak-uniformhypergraphisoneinwhich
X
each edge has size k. Some view k-uniform hypergraphs as the most natural generalization of a graph
r (agraphisa2-uniformhypergraph)andonemightalsogeneralizetheforbiddensubgraphproblemtoa
a
forbidden k-uniform subhypergraph problem. There are both asymptotic results and exact bounds (e.g.
[dCF00],[Pik08],[Fu¨r91]).
Forbidden Configurations is a different (but also natural) generalization that is studied mainly by
Richard Anstee and his colaborators. We consider the following problem: Given m ∈ N and a hy-
pergraph F, find the maximum number of edges in a simple hypergraph H on m vertices that avoids
havingasubhypergraphisomorphictoF.
†Email:[email protected],UNAM
subm.toDMTCS (cid:13)c bytheauthorsDiscreteMathematicsandTheoreticalComputerScience(DMTCS),Nancy,France
2 MiguelRaggi
Wefinditconvenienttousethelanguageofmatricestodescribehypergraphs:Eachcolumnofa{0,1}-
matrixcanbeviewedasanincidencevectoronthesetofrows.
Definition 1.1. Define a matrix to be simple if it is a {0,1}-matrix with no repeated columns. Then
an m × n simple matrix corresponds to a simple hypergraph (or set system) on m vertices with n
distinctedgeswhereweallowtheemptyedge. Let(cid:107)A(cid:107)denotethenumberofcolumnsinA(whichisthe
cardinalityoftheassociatedsetsystem).
Definition1.2. LetAandBbe{0,1}-matriceswiththesamenumberofrows.Definetheconcatenation
[A|B] to be the configuration that results from taking all columns of A together with all columns of B.
Fort∈N,wedefinetheproduct
t·A:=[A|A| ··· |A].
(cid:124) (cid:123)(cid:122) (cid:125)
ttimes
Ourobjectsofstudyare{0,1}-matriceswithrowandcolumnorderinformationstrippedfromthem:
Definition1.3. Two{0,1}-matricesaresaidtobeequivalentifoneisarowandcolumnpermutationof
theother. Thisdefinesanequivalencerelation. Anequivalenceclassiscalledaconfiguration.
Abusingnotation,wewillcommonlyusematrices(representatives)andtheircorrespondingconfigura-
tionsinterchangeably.
Definition 1.4. For a configuration F and a {0,1}-matrix A (or a configuration A), we say that F is a
subconfigurationofA,andwriteF ≺AifthereisarepresentativeofF whichisasubmatrixofA. We
sayAhasnoconfigurationF (ordoesn’tcontainF asaconfiguration)ifF isnotasubconfiguration
ofA. LetAvoid(m,F)denotethesetofallm-rowedsimplematriceswithnosubconfigurationF.
Ourmainextremalproblemistocompute
forb(m,F)=max{(cid:107)A(cid:107) : A∈Avoid(m,F)}.
A
Perhapssomeexamplesareuseful:
(cid:20) (cid:21)
1
• forb(m, )=m+1,sincewecantakeallcolumnswithatmostone1.
1
(cid:20) (cid:21)
1
• forb(m, )=2,sincewemayonlytakethecolumnof1’sandthecolumnof0’s(i.e. theempty
0
setandthecompleteset).
(cid:20) (cid:21)
• forb(m, 1 1 )=(cid:0)m(cid:1)+(cid:0)m(cid:1)+(cid:0)m(cid:1),bytakingallcolumnswithatmosttwo1’s. Theproofthat
1 1 2 1 0
thisisindeedthemaximumiseasyandcanbefoundin[Rag11].
Surveysonthetopiccanbefoundin[Ans]and[Rag11]. LetAc denotethe{0,1}-complementofA
(replaceevery0inAbya1andevery1bya0). Notethatforb(m,F)=forb(m,Fc).
Remark1.5. LetF andGbeconfigurationssuchthatF ≺G. Thenforb(m,F)≤forb(m,G).
FindingthenumberpredictedbytheAnstee-SaliConjectureisNP-hard 3
Wesayacolumnαhascolumnsumtifithasexactlytones. Define0 tobeacolumnwithm0’s
m
and1 tobeacolumnofm1’s.
m
Forasetofrows S, weletA| denotethesubmatrixof A givenbyrestrictingtherows ofA toonly
S
thoseinS.
AnimportantgeneralresultduetoFu¨rediappliestosimpleortonon-simpleconfigurations.
Theorem1.6(Z.Fu¨redi). LetF beagivenk-rowed{0,1}−matrix. Thenforb(m,F)isO(mk).
We desire more accurate asymptotic bounds. Anstee and Sali conjectured that the best asymptotic
boundscanbeachievedwithcertainproductconstructions.
Definition1.7. LetAandBbe{0,1}-matrices. WedefinetheproductA×Bbytakingeachcolumnof
AandputtingitontopofeverycolumnofB. Hereisanexampleofaproduct:
 
0 0 1 1 1 1
(cid:20) (cid:21) (cid:20) (cid:21) A
0 1 1 1 0  0 0 0 0 1 1 
A= , B = =⇒ ×= .
0 0 1 0 1  1 0 1 0 1 0 
B
0 1 0 1 0 1
Notethatthisisawelldefinedoperationinconfigurations.
We are interested in asymptotic bounds for forb(m,F). Let I be the m×m identity matrix, Ic
m m
be the {0,1}-complement of I (all ones except for the diagonal) and let T be the tower matrix: a
m m
matrixcorrespondingtoamaximumchaininthepartiallyorderedsetofthepowersetofthevertices. For
example,
 
0 1 1 1 1
0 0 1 1 1
T4 =0 0 0 1 1
0 0 0 0 1
AnsteeandSaliconjecturedthattheasymptotically“best”constructionsavoidingasingleconfiguration
wouldbeproductsofI,IcandT.
Conjecture1.8. [AS05]LetF beaconfiguration. Let
P (a,b,c):=I ×...×I ×Ic×...×Ic×T ×...×T ,
r r r r r r r
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
atimes btimes ctimes
DefineX(F)tobethelargestnumbersuchthatthereexistnumbersa,b,c∈Nwitha+b+c=X(F)
suchthatforallr ∈N,
F ⊀P (a,b,c).
r
Thenforb(m,F)isΘ(mX(F)).
Observe that X(F) is always an integer. Also note that (cid:107)P (a,b,c)(cid:107) = ra+b · (r + 1)c which is
r
Θ(rX(F)),sobytakingr = (cid:100)m/X(F)(cid:101)(andperhapsdeletingsomerowsincaseX(F) (cid:45) m),wehave
that(cid:107)P (a,b,c)(cid:107)isΩ(mX(F)),sothefactthatforb(m,F)isΩ(mX(F))isbuiltintotheconjecture. In
r
ordertoprovetheconjecture,allthatwouldberequiredwouldbetoprovethatforb(m,F)isO(mX(F))
foreveryF. Adisproofcouldbepotentiallyeasier,asonlyacounterexamplewouldberequired.
4 MiguelRaggi
Theconjecturehasbeenprovenforallk×(cid:96)configurationsF withk ∈{1,2,3}andmanyotherscases
invariouspapers. Theproofsfork = 2arein[AGS97], fork = 3in[AGS97], [AFS01], [AS05]. For
(cid:96) = 2,theconjecturewasverifiedin[AK06]. Fork = 4,allcaseseitherwhentheconjecturepredictsa
cubicboundforF orwhenF issimplewerecompletedin[AF10]. Fork = 4andF non-simple,there
arethreeboundarycaseswithquadraticbounds,oneofwhichisestablishedin[ARS12]. Fork ∈ {5,6}
someresultscanbefoundin[ARS11].
AnsteehaslongconjecturedthatevenfindingX(F)givenF wasnotatrivialtask,andthequestionof
itsNP-hardnesswaslongconjecturedbutnotproven([Ans],[Rag11]). Inthispaperwesettlethatfinding
X isindeedNP-hardandwealsonotethatoneofthedecisionversionsassociatedwiththisoptimization
problemisNP-complete,addingthisfunctiontothelonglistoffunctionsknowntobeNP-complete,with
theinterestingplusthatthisfunctionisconjecturedtogivetheexponentoftheasymptoticgrowthofforb.
For relatively small configurations F we have a computer program that yields the answer (relatively)
quickly. Thesourcecode(inC++)canbefreelydownloadedfrom:
http://matmor.unam.mx/˜mraggi/
This program can compute X(F) for F having less than ∼10 rows in just a few minutes. This task
takesmerelyexponentialtime,notdoublyexponential(asitisoftenthecasewithforbiddenconfiguration
problems).ThisprogramwaswrittentoperformmanyothertasksotherthanfindingX(F).Adescription
ofthealgorithmusedforthistaskisintheappendix4.
2 Results
TherearetwonaturaldecisionproblemsassociatedwithX(F): GivenF andkasinputs,
1. IsittruethatX(F)<k?
2. IsittruethatX(F)≥k?
WeprovethatthefirstofthetwodecisionproblemsisinNPbyexhibitingacertificatewhichcanbe
checkedinpolynomialtime.
Themainresultofthispaperisthefollowing:
Theorem 2.1. Finding X(F) is an NP-hard problem. In other words, should a polynomial-time algo-
rithm exist for finding X(F) given F, then every problem in NP could be solved in polynomial time.
Furthermore,theproblemof“givenF andk,isX(F)<k?” isinNP.
Beforeprovingthistheorem,weneedafewlemmas.
Observation2.2. LetF beaconfigurationwithnrows. ThenX(F) ≤ n. Indeed,assumeforthesake
ofcontradictionthata,b,andcaresuchthata+b+c=n+1andF ⊀P (a,b,c). Wemayplaceeach
r
rowofF eachintoadifferentfactoroftheproduct. TheextramatrixensurestherepeatedcolumnsofF
getrepeatedasmanytimesasneeded.
This observation in particular implies that if a polynomial time algorithm existed for any of the two
decisionversionsoftheproblem,thenwe’dhaveapolynomialtimealgorithmforfindingX(F),which
togetherwithTheorem2.1wouldmakethedecisionversionoffindingX(F)anNP-completeproblem.
Asimple(butsurprising)corollaryofconjecture1.8,ifitweretrue,wouldbethatrepeatingcolumns
morethantwiceinF hasnoeffectontheasymptoticbehaviorofforb(m,F). Inotherwords,assuming
FindingthenumberpredictedbytheAnstee-SaliConjectureisNP-hard 5
theconjectureweretrue,themultiplicityofacolumninaconfigurationwouldnotaffecttheasymptotic
boundand,asymptotically,itwouldonlymatterifacolumnisnotthere(hasmultiplicity0),appearsonce
(hasmultiplicity1),orappears“multipletimes”(hasmultiplicity2ormore). Formally,
Proposition 2.3. Let F = [G|t · H] with G and H simple {0,1}-matrices that have no columns in
t
common. ThenX(F )=X(F )forallt≥2. Inparticular,iftheconjectureweretrue,thenforb(m,F )
2 t t
andforb(m,F )wouldhavethesameasymptoticbehavior.
2
Proof: Itsufficestoshowthatgivent,G,H,a,bandc,thereexistsanRsuchthatforeveryr ≥ R,we
have
F =[G|2·H]≺P (a,b,c) ⇐⇒ F =[G|t·H]≺P (a,b,c).
2 r t r
Since F ≺ F , we only need to prove that if F ≺ P (a,b,c) for some r, then F ≺ P (a,b,c)
2 t 2 r t R
for some R. Suppose then F is contained in the product P (a,b,c) for some r. The idea is to find a
2 r
subconfigurationofP (a,b,c)inwhichtherearesomecolumnswithmultiplicity1,andforthecolumns
r
withmultiplicity2ormore,themultiplicitydependsonr. Weneedrlargeenoughsothatthemultiplicity
ofanyonecolumn(withmultiplicityof2ormore)islargerthant. LetxbethenumberofrowsofF .
t
Noticethefollowingthreefacts,whichincludedefinitionsforE ,E andE .
I Ic T
E (x,r):=[(r−x)·0 |I ]≺I
I x x r
E (x,r):=[(r−x)·1 |Ic]≺Ic
Ic x x r
(cid:106)r(cid:107)
E (x,r):= ·T ≺T .
T x x r
The first and second facts are easy to see; just take any subset of x rows from I or Ic. The third
r r
statementistruebytakingthe(cid:98)r/x(cid:99)-throwofT ,the2(cid:98)r/x(cid:99)-throwofT ,etcetera,uptothex(cid:98)r/x(cid:99)-th
r r
row. Forexample,ifr =5andx=2,wemaytakethesecondandfourthrowfromT :
5
 
0 1 1 1 1 1
0 0 1 1 1 1 (cid:20) (cid:21)
  0 0 1 1 1 1
T5 =0 0 0 1 1 1 =⇒ T5|{2,4} = 0 0 0 0 1 1 =ET(2,5)
0 0 0 0 1 1
0 0 0 0 0 1
Note that in the three configurations E (x,r),E (x,r) and E (x,r), we have that there are some
I Ic T
columnsofmultiplicity1andtherearesomecolumnsforwhichtheirmultiplicitycanbemadeaslarge
as we wish by making r large. Formally, let E(x,r) be one of E (x,r) or E (x,r) or E (x,r). We
I Ic T
havethatforeveryx-rowedcolumnαtherearethreepossibilities: eitherλ(α,E(x,r)) = 0forallr,or
λ(α,E(x,r))=1forallr,or lim λ(α,E(x,r))=∞.
r→∞
If α is a column for which lim λ(α,E(x,r)) = ∞, we may conclude that there is an R for which
r→∞
λ(α,E(x,r))≥tforeveryr ≥R.
Since F is contained in P (a,b,c) for some r, the columns in H will have multiplicity at least 2 in
2 r
somesubsetoftherowsofP (a,b,c). WeseethatF isalsoasubconfigurationofP (a,b,c).
r t R
6 MiguelRaggi
3 Proof of the Main Theorem
Wenowprovethemaintheorem.
Proof: First we prove that the decision problem has a certificate which can be checked in polynomial
time. AcertificatethatindeedX(F) < k wouldhavetobeaproofthatF ≺ P (a,b,c)foreachtriple
r
a,b,c for which a+b+c = k. Note that there are at most a quadratic (with respect to the number of
rows)numberofa,b,c’swhichsatisfytheequation,sincethequestionhasatrivial“yes”answerwhenk
morethanthenumberofrows(Observation2.2).
GivenF andAconfigurations,onecaneasilyconstructacertificatethataconfigurationF isindeeda
subconfigurationofaconfigurationA: explicitlystatewhichpermutationofF appearsinexactlywhich
rowsandcolumnsofA. ForthecaseA=P (a,b,c),acertificateonlyneedstospecifywhichrowsofF
r
go inside which factors, so at most a quadratic number of these certificates-of-being-a-subconfiguration
suffice.
WenowprovethatfindingX(F)isNP-hard. Supposethereexistedsomepolynomial-timealgorithm
that finds X(F) given F. We shall prove that there would then exist a polynomial time algorithm for
GRAPH COLORING. Suppose we are given a graph G and we wish to find the minimum number of
colorsforwhichthereexistsagoodcoloringofthegraph. Wemayassumenoisolatedvertices.
Theideaistoconstructa3-partmatrixF(G)inwhichthefirsttwopartsensurethereisnoT orIcina
maximumproductoftheformP (a,b,c)withnosubconfigurationF(G),andthelastpartisconstructed
r
so that a partition into I(cid:48)s produces a partition of the vertices of the graph into independent sets and
vice-versa.
Suppose G has n vertices and e edges. Let M be a large number with M ≥ n+2 and let S be the
incidence matrix of G (i.e., the edges of G are encoded as columns with two 1’s corresponding to the
verticesthatbelongtotheedge). Constructthefollowingsimplematrix:
 1 0 Ic 
M M
F(G):= 1 TM 
S 0
ClearlywecanconstructF(G)inpolynomialtime(withrespecttothenumberofverticesofG). We
provenowthatwehaveχ(G)=X(F)−2M+1,whichinturnwouldyieldapolynomialtimealgorithm
forGRAPHCOLORING,providedwehadapolynomialtimealgorithmforX(F).
Now,letusstudythepossibilitiesforaproductoftypeP (a,b,c)thatdoesnothaveF(G)asasubcon-
r
figurationforanyr. Ifb(cid:54)=0,thenwecouldplaceallof[1|Ic ]intheIc partofP (a,b,c),soa+b+c
M r
wouldbeatmost1+M +n(usingLemma2.2). Thesameistruewhenc(cid:54)=0. Butifweletb=c=0,
P (a,0,0)isjustaproductofI’s,soletuscalculatehowmanyI(cid:48)swecanmultiplytogetherandstillnot
r
createasubconfigurationF(G).
InorderforF(G)tobeapartofaproductofI(cid:48)s,everyrowof[1|Ic ]and[1|T ]mustbeinaseparate
M M
(cid:20) (cid:21)
1
factorI, sincethereisno inI (andalsoseparatefromtherowsof[S|0], sinceweareassumingG
1
hasnoisolatedvertices).
(cid:20) (cid:21)
1
Then two rows of the [S|0] part can be in the same I if and only if there is no in those two
1
rows, which, intermsofthegraph, meansthereisnoedgebetweenthosetwovertices. Inotherwords,
partitioning [S|0] into I’s is equivalent to partitioning the vertices of G into independent sets. So if the
FindingthenumberpredictedbytheAnstee-SaliConjectureisNP-hard 7
graphGcannotbecoloredwithχ(G)−1colorsandthisisthemaximum,thismeansthatX(F)=a=
2M +χ(G)−1≥n+M +1. Thenχ(G)=X(F(G))−2M +1.
References
[AF10] R.P. Anstee and B. Fleming, Two refinements of the bound of Sauer, Perles and Shelah and
VapnikandChervonenkis,DiscreteMathematics310(2010),3318–3323.
[AFS01] R.P.Anstee,R.Ferguson,andA.Sali,SmallforbiddenconfigurationsII,ElectronicJournalof
Combinatorics8(2001),R425pp.
[AGS97] R.P. Anstee, J.R. Griggs, and A. Sali, Small forbidden configurations, Graphs and Combina-
torics13(1997),97–118.
[AK06] R.P. Anstee and P. Keevash, Pairwise intersections and forbidden configurations, European
JournalofCombinatorics27(2006),1235–1248.
[Ans] R.P.Anstee,Asurveyofforbiddenconfigurationresults,http://www.math.ubc.ca/∼anstee/.
[ARS11] R.P. Anstee, Miguel Raggi, and Attila Sali, Forbidden configurations: Quadratic bounds,
preprint.(2011).
[ARS12] R.P. Anstee, M. Raggi, and A. Sali, Evidence for a forbidden configuration conjecture: One
morecasesolved,DiscreteMathematics312(2012),no.17,2720–2729.
[AS05] R.P.AnsteeandA.Sali,SmallforbiddenconfigurationsIV,Combinatorica25(2005),503–518.
[dCF00] DominiquedeCaenandZ.Fu¨redi,Themaximumsizeof3-uniformhypergraphsnotcontaining
aFanoplane,JournalofCombinatorialTheory,SeriesB78(2000),274–276.
[ES46] P. Erdo˝s and A.H. Stone, On the structure of linear graphs, Bulletin of the American Mathe-
maticalSociety52(1946),1089–1091.
[ES66] PaulErdo˝sandMiklo´sSimonovits,Alimittheoremingraphtheory.,StudiaScientiarumMath-
ematicarumHungarica1(1966),51–57.
[Fu¨r91] Z. Fu¨redi, Tura´n type problems, Surveys in Combinatorics (Proc. of the 13th British Combi-
natorialConference),ed.A.D.Keedwell,CambridgeUniv.Press.LondonMath.Soc.Lecture
NoteSeries166(1991),253–300.
[Pik08] O. Pikhurko, An exact bound for Tura´n result for the generalized triangle, Combinatorica 28
(2008),187–208.
[Rag11] MiguelRaggi,Forbiddenconfigurations,Ph.D.thesis,UniversityofBritishColumbia,2011.
8 MiguelRaggi
4 Appendix: Algorithm to find X(F)
In this section we describe the algorithm used by the software described in the introduction. It runs
in exponential time, of course, but it has been helpful for finding boundary configurations with given
asymptotic bounds (see [ARS11], [Rag11] and [Ans]). Perhaps this program might be used to find a
counterexampletotheAnstee-Saliconjecture,providedoneexists,anditisn’tverylarge.
4.1 Representation of a Configuration
Weareinterestedinanefficientrepresentationforconfigurationsinordertoperformthetasksdescribed
above. Intheprogressofourinvestigations,wehavehadvariousversionsoftheprogram.
Mostofwhatwewanttheprogramtodoinvolvesperformingahugenumberofconfigurationcompar-
isonoperations,whichistestingwhetherornotaconfigurationF isasubconfigurationofaconfiguration
A. Asafirstapproachitwouldseemasif,forthistask,wewouldberequiredtotesteachrowandcolumn
permutationofF againsteachsubmatrixofA.Thisisofcourseaveryslowwaytodothis.Asimpletrick
tospeedupthecomputationsistokeepthecolumnsofaconfigurationalwaysinsomecanonicalorder.
Then,totestwhetherornotaconfigurationF iscontainedinanotherA,wejustneedtopermuterowsof
F andtakesubsetsS ofrowsofAandplacethecolumnsofA| incanonicalorder.
S
Mostofthetaskswewishthisprogramtoperforminvolvecheckingwhetherornotagiven(fixed)con-
figurationF isasubconfigurationofavastnumberofconfigurationsA. Inparticular,anypre-processing
wedoonF canbeconsideredasalmostfree. Forexample,findingallrowpermutationsofF andstoring
themwouldneedtobedoneonceforeachconfigurationF,andnotatallforconfigurationsA.
Aftermanyattemptsandexperiments,itseemedthatthebest(fastest)waytostoreaconfigurationthat
mademanyoftheothertasksreasonablyfastisthis:Maintainanarrayofintegerswheretheindicesofthe
array,writteninbinary,arethecolumnsoftheconfiguration,andtheactualnumbersofthearrayrepresent
thenumberoftimesacolumnappears. Thatistosay,aconfigurationF inmrowsisrepresentedbyan
array(C++vector) Fofsize2m. Foranumberα, considerthebinaryrepresentationof α andconsider
itasacolumnwithmrows. Ifnecessary,putenough0’satthebeginningofthebinaryrepresentationin
ordertohavetherequiredmbits. ThenumberF[α](theα-thnumberofthearray)representsthenumber
oftimesthatcolumnαappearsinconfigurationF. Intheimplementation, weuseanarrayofunsigned
charactersinsteadofintegers,sinceweneverneedaconfigurationwiththesamecolumnrepeatedmore
than255times. Anunsignedcharacterconsistsof1byte(8bits).
Forexample,thearrayF = [1,0,0,2,0,1,0,1]representsthefollowingconfiguration(noticeithas3
rows,sincethearrayhassize8=23):
 
0 0 0 1 1
F =0 1 1 0 1.
0 1 1 1 1
To see this, remember we start from 0. There is a one in position 0 = 000 , meaning the colum
b
(0,0,0)T getsrepeatedonetime. Atwoinposition3 = 011 , aoneinposition5 = 101 andaonein
b b
position7=111 . Thecolumnsofthismatrixaretherepresentationsofthesenumbersinbinaryform.
b
Anobservantreadermightcomplainthatthishasthedisadvantagethatitrequiresstoring2mbytes,and
ifF doesn’thavemanycolumns,mostofthosewillbe0’s. Butit’saminordisadvantage,becauseeven
at10rowswewouldonlyneed1024bytes,andweusuallyhaveconfigurationsforwhichthenumberof
rowsis5orless(32bytes). Perhapsthiswouldbecomemoreofanissuewithconfigurationswithahigh
FindingthenumberpredictedbytheAnstee-SaliConjectureisNP-hard 9
numberofrows,butforthoseconfigurations,mostofourtaskswouldrequiretoomuchtimetobeofany
practicaluseanyway.
Wecametothisrepresentationafteranimplementationwhichrepresentedcolumnsasanarrayofbits
(C++bitset)andstoringthemintoanorderedtree-likestructure(C++multisetfromtheSTL).Thismight
beamorenaturalimplementation,butprofilingthecodemadeclearthattheprogramwasspendingmostof
itstimecountinghowmanycolumnsofacertaintypeappearedinaconfiguration,andwasalsospending
a considerable amount of time navigating the tree. Explicitly storing the number of times each column
appears,andmakingthatnumberinstantlyaccessiblebystoringitinanarray(forrandomaccess)givesa
verynoticeablespeedupandallowsustoconsiderlargerproblems. Byrepresentingcolumnsasnumbers,
wecandoalotofpreprocessingandcomputelargetablesinwhichwehavealmostinstantaccesstime.
Forexample, considerthefollowingproblem, whichhastobedonemanytimesforourtasks: Given
acolumnαandasubsetS (representedbyanintegeraswritteninbinary),whatcolumnisα ? Thisis
S
relativelyslowtocompute,butwecanfilloutatablebypreprocessingtospeedupanyfurtheraccessto
it. Sincewedothisafewmilliontimes,theinvestmentissound.
Theotheradvantageisthatitbecomesimmediatelyclearhowtocomparetwoconfigurationswiththe
samenumberofrowstoseeifoneisacolumn-permutationsubmatrixoftheother;checkifforanycolumn
(index)theintegeratpositioncofthefirstarrayisbiggerthanthatofthesecondarray. TocheckifF is
asubconfigurationofA,wewouldneedtofindallpermutationsoftherowsofF (whichweneedtodo
justonceperconfiguration).
4.2 Subconfigurations
SupposeF andAareconfigurationsandwewanttodecideifF ≺ A. Thenforeverys-tupleofrowsS
(wheresisthenumberofrowsofF),wecanextractfromAtheconfigurationA| easilywithourpre-
S
storedtableofcolumnsandsubsets. Oncewe’vedonethisforeachcolumnofAandfoundA| ,thenfor
S
eachpermutationofrowsofF,wecheckifeverycolumnαinthearraycorrespondingtoconfigurationF
appearslessthanorequaltothecorrespondingnumberforcolumnαinthearrayofA| . Wecancheck
S
everysubsetS likethis. Ifatanypointthisisso,wecanreturntrue.
There are a few speedups. Sometimes it’s immediately obvious a configuration can’t be contained in
another. Forexample,iftherearemore1’sinF thaninA,orifF hasmorecolumnsorrowsthanA,then
F ⊀A.
4.3 Determining X(F)
GivenaconfigurationF,wewishtofindX(F). Inotherwords,wewishtofindtheconjecturedasymp-
toticboundforforb(m,F).
WemaymakeasimplificationusingProposition2.3andassumethemultiplicityofanycolumnofF is
atmost2.
First,supposewewantedtotestwhetherornotthereexistsrsuchthatconfigurationF iscontainedin
theproduct
P (a,b,c)=I ×...×I ×Ic×...×Ic×T ×...×T .
r r r r r r r
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
atimes btimes ctimes
Buildingthisobjectwithr = RascalculatedinProposition2.3wouldbeprohibitivelyslow. Instead,
we build a set X of subconfigurations from P (a,b,c) such that if F ≺ P (a,b,c), then F ≺ X for
R R
someX ∈X.
10 MiguelRaggi
Recall that if F ≺ P (a,b,c), then the rows of F get partitioned into a+b+c parts (a part can be
R
empty), where each part belongs to a factor of the product P (a,b,c). Because of Proposition 2.3, we
R
canassumeeachcolumnappearsatmosttwice. Givens∈N,considerthefollowingmatrices:
A (s)=(cid:2)t·0 I (cid:3) , A (s)=(cid:2)t·1 Ic(cid:3) , A (s)=t·(cid:2)T (cid:3).
I s s Ic s s T s
Weseethatans-rowedconfigurationF (witheachcolumnrepeatedatmostttimes)iscontainedinI
m
forsomelargem,ifandonlyifF ≺ A (s). WecanthenconsiderallpartitionsofrowsofF andseeif
I
eachpartiscontainedinthecorrespondingA ,A orA .
I Ic T
Forexample,totestwhether
 
1 0 1 1
0 1 0 1
 
F =1 1 0 0
 
0 0 1 1
1 0 0 1
iscontainedinI ×T ×T,wewouldpartitiontherowsofF inthreeparts. Inthiscase,F hasfiverows,
soconsider,forexample,thefollowingpartitionof5: (2,2,1). Considerthefollowingrepresentatives:
(cid:20) (cid:20) (cid:21) (cid:12)(cid:20) (cid:21)(cid:21) (cid:20) (cid:21)
0 (cid:12) 1 0 0 1 1 (cid:2) (cid:3)
AI(2)= s· 0 (cid:12)(cid:12) 0 1 , AT(2)=s· 0 0 1 , AT(1)=s· 0 1 ,
andbuilda5-rowedmatrixA := A (2)×A (2)×A (1). IfF ≺ A,thenF ≺ I ×T ×T. Ifwedo
I T T
thisforeverypossiblepartitionof5,wegetthedesiredresult.
To find X(F) we can build a tree of possibilities of products and prune whenever we hit a node that
alreadyhasconfigurationF:
We return the product with the largest number of factors for which F is not a subconfiguration of a
product of this form, observing that if F is a subconfiguration of a product P (a,b,c), then it will be a
r
subconfigurationinanyproductP (a(cid:48),b(cid:48),c(cid:48))witha(cid:48) ≥a,b(cid:48) ≥bandc(cid:48) ≥c.
r
4.4 Finding Boundary Cases and Classifying Configurations
Givennumberofrowssandanumberk,wewishtofindallboundarycases,thatismaximalandminimal
F suchthatX(F) = k. WemakeuseoftheprogramdescribedintheprevioussectiontofindX(F)for
manydifferentF. Theboundarycasesforsmallsandkcanbefoundin[Ans]and[Rag11].