Discovering Archipelagos of Tractability for Constraint Satisfaction and Counting

28 Pages·2017·0.62 MB·English

Discovering Archipelagos of Tractability for
Constraint Satisfaction and Counting
∗
RobertGanian M.S.Ramanujan StefanSzeider
ViennaUniversityofTechnology UniversityofBergen ViennaUniversityofTechnology
Vienna,Austria Bergen,Norway Vienna,Austria
[email protected] [email protected] [email protected]
5
1
0
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l
u Abstract
J
TheConstraintSatisfactionProblem(CSP)isacentralandgenericcomputationalproblemwhich
0
providesacommonframeworkformanytheoreticalandpracticalapplications.Acentrallineofresearchis
2
concernedwiththeidentificationofclassesofinstancesforwhichCSPcanbesolvedinpolynomialtime;

] suchclassesareoftencalled“islandsoftractability.” Aprominentwayofdefiningislandsoftractability
S
for CSP is to restrict the relations that may occur in the constraints to a fixed set, called a constraint
D
language, whereas a constraint language is conservative if it contains all unary relations. Schaefer’s
.
s famous Dichotomy Theorem (STOC 1978) identifies all islands of tractability in terms of tractable
c
constraintlanguagesoveraBooleandomainofvalues. Sincethenmanyextensionsandgeneralizationsof
[
thisresulthavebeenobtained.Recently,Bulatov(TOCL2011,JACM2013)gaveafullcharacterizationof

2 allislandsoftractabilityforCSPandthecountingversion#CSPthataredefinedintermsofconservative
v
constraintlanguages.
9
ThispaperaddressesthegenerallimitofthementionedtractabilityresultsforCSPand#CSP,that
7
4 theyonlyapplytoinstanceswhereallconstraintsbelongtoasingletractablelanguage(ingeneral,the
2 unionoftwotractablelanguagesisn’ttractable). Weshowthatwecanovercomethislimitationaslongas
0 wekeepsomecontrolofhowconstraintsoverthevariousconsideredtractablelanguagesinteractwith
.
7 eachother. ForthispurposeweutilizethenotionofastrongbackdoorofaCSPinstance,asintroduced
0 byWilliamsetal.(IJCAI2003),whichisasetofvariablesthatwheninstantiatedmovestheinstanceto
5 anislandoftractability,i.e.,toatractableclassofinstances. Weconsiderstrongbackdoorsintoscattered
1
classes,consistingofCSPinstanceswhereeachconnectedcomponentbelongsentirelytosomeclass
:
v from a list of tractable classes. Figuratively speaking, a scattered class constitutes an archipelago of
Xi tractability. Themaindifficultyliesinfindingastrongbackdoorofgivensizek;onceitisfound,we
cantryallpossibleinstantiationsofthebackdoorvariablesandapplythepolynomialtimealgorithms
r
a associatedwiththeislandsoftractabilityonthelistcomponentwise. Ourmainresultisanalgorithm
that,givenaCSPinstancewithnvariables,findsintimef(k)nO(1)astrongbackdoorintoascattered
class(associatedwithalistoffiniteconservativeconstraintlanguages)ofsizekorcorrectlydecidesthat
thereisn’tsuchabackdoor. Thisalsogivestherunningtimeforsolving(#)CSP,providedthat(#)CSPis
polynomial-timetractablefortheconsideredconstraintlanguages. Ourresultmakessignificantprogress
towards the main goal of the backdoor-based approach to CSPs – the identification of maximal base
classesforwhichsmallbackdoorscanbedetectedefficiently.
∗ResearchsupportedbytheAustrianScienceFunds(FWF),projectP26696X-TRACT.
1 Introduction
TheConstraintSatisfactionProblem(CSP)isacentralandgenericcomputationalproblemwhichprovidesa
common framework for many theoretical andpractical applications [24]. An instanceof CSP consists of
acollectionofvariablesthatmustbeassignedvaluessubjecttoconstraints,whereeachconstraintisgiven
intermsofarelationwhosetuplesspecifytheallowedcombinationsofvaluesforspecifiedvariables. The
problemwasoriginallyformulatedbyMontanari[32],andhasbeenfoundequivalenttothehomomorphism
problemforrelationalstructures[18]andtheproblemofevaluatingconjunctivequeriesondatabases[27].
In general CSP is NP-complete. A central line of research is concerned with the identification of classes
of instances for which CSP can be solved in polynomial time. Such classes are often called “islands of
tractability”[27,28].
A prominent way of defining islands of tractability for CSP is to restrict the relations that may occur
in the constraints to a fixed set Γ, called a constraint language. A finite constraint language is tractable
if CSP restricted to instances using only relations from Γ, denoted CSP(Γ), can be solved in polynomial
time. Schaefer’sfamousDichotomyTheorem[38]identifiesallislandsoftractabilityintermsoftractable
constraint languages over the two-element domain. Since then, many extensions and generalizations of
thisresulthavebeenobtained[26,12,29,39]. TheDichotomyConjectureofFederandVardi[17]claims
that for every finite constraint language Γ, CSP(Γ) is either NP-complete or solvable in polynomial time.
Schaefer’sDichotomyTheoremshowsthattheconjectureholdsfortwo-elementdomains;morerecently,
Bulatov [3] showed the conjecture to be true for three-element domains. Several papers are devoted to
identifyingconstraintlanguagesΓforwhichcountingCSP,denoted#CSP(Γ),canbesolvedinpolynomial
time[5,13,6],i.e.,wherethenumberofsatisfyingassignmentscanbecomputedinpolynomialtime. Such
languagesΓarecalled#-tractable.
Aconstraintlanguageover isconservativeifitcontainsallpossibleunaryconstraintsover ,anditis
D D
semi-conservativeifitcontainsallpossibleunaryconstantconstraints(i.e.,constraintsthatfixavariabletoa
specificdomainelement). Thesepropertiesofconstraintlanguagesareverynatural,asonewouldexpect
inpracticalsettingsthattheunaryrelationsarepresent. Indeed,someauthors(e.g.,[10])evendefineCSP
sothateveryvariablecanhaveitsownsetofdomainvalues,makingconservativenessabuilt-inproperty.
Recently,Bulatov[4]gaveafullcharacterizationofalltractableconservativeconstraintlanguagesoverfinite
domains. Furthermore,Bulatov[5]gaveafullcharacterizationofall#-tractableconstraintlanguagesover
finitedomains. Thus,Bulatov’sresultsidentifyallislandsof(#-)tractabilityoverfinitedomainswhichcanbe
definedintermsofaconservativeconstraintlanguage.
A general limit of tractability results for CSP and #CSP based on constraint languages, such as the
mentionedresultsofSchaeferandBulatov,isthattheyonlyapplytoinstanceswhereallconstraintsbelongto
asingletractablelanguage. Onecannotarbitrarilycombineconstraintsfromtwoormoretractablelanguages,
as in general, the union of two tractable languages isn’t tractable (see Section 2). In this paper we show
thatwecanovercomethislimitationaslongasconstraintsoverthevariousconsideredtractablelanguages
interactwitheachotherinacontrolledmanner. Forthispurposeweutilizethenotionofastrongbackdoor
ofaCSPinstance,asintroducedbyWilliamsetal.[40]. AsetB ofvariablesofaCSPinstanceisastrong
backdoorintoatractableclass ifforallinstantiationsofthevariablesinB,thereducedinstancebelongs
H
to . Inthispaper,weconsiderstrongbackdoorsintoascatteredclass,denoted ,consisting
1 d
H H ⊕···⊕H
ofallCSPinstancesIsuchthateachconnectedcomponentofIbelongsentirelytosomeclassfromalistof
tractableclasses ,..., . Figurativelyspeaking, ... constitutesanarchipelagooftractability,
1 d 1 d
H H H ⊕ ⊕H
consistingoftheislands ,..., .
1 d
H H
2
Ourmainresultisthefollowing:
Theorem1. LetΓ ,...,Γ besemi-conservativefiniteconstraintlanguagesoverdomain ,andlet ∗ be
1 d
D D
thelanguagecontainingallrelationsover . IfΓ ,...,Γ aretractable(or#-tractable),thenCSP( ∗)(or
1 d
D D
#CSP( ∗),respectively)canbesolvedintime22O(k) nO(1) forinstanceswithnvariablesthathaveastrong
D ·
backdoorofsizek intoCSP(Γ ) ... CSP(Γ ).
1 d
⊕ ⊕
NotethattherearenaturalCSPinstanceswhichhaveasmallstrongbackdoorintothescatteredclass
CSP(Γ ) ... CSP(Γ )butrequirestrongbackdoorsofarbitrarilylargesizeintoeachindividualbase
1 d
⊕ ⊕
classCSP(Γ ). Thepowerofastrongbackdoorintoascatteredclassoveroneintoasingleclassstemsfrom
i
thefactthattheinstantiationofvariablesinthebackdoorcanservetwopurposes. Thefirstistoseparate
constraintsintocomponents,eachbelongingentirelytosomeCSP(Γ )(possiblyevendifferentCSP(Γ )’s
i i
fordifferentinstantiations),andthesecondistomodifyconstraintssothatoncemodified,thecomponent
containingtheseconstraintsbelongstosomeCSP(Γ ).
i
Whenusingthebackdoor-basedapproach,themaincomputationaldifficultyisindetectingsmallbackdoor
setsintothechosenbaseclass. Thistaskbecomessignificantlyharderwhenthebaseclassesaremademore
general. However,weshowthatwhilescatteredclassesaresignificantlymoregeneralthansinglebaseclasses,
wecanstilldetectstrongbackdoorsintosuchclassesinFPTtime. Theformalstatementofthisresult,which
representsourmaintechnicalcontribution,isthefollowing.
Lemma1. Thereisanalgorithmthat,givenaCSPinstanceIandaparameterk,runsintime22O(k) nO(1)
·
andeitherfindsastrongbackdoorofsizeatmostk inIintoCSP(Γ∗) ... CSP(Γ∗)orcorrectlydecides
1 ⊕ ⊕ d
thatnoneexists.
Here Γ∗ Γ is obtained from Γ by taking the closure under partial assignments and by adding a
i ⊇ i i
redundantrelation.
Weremarkthatthefinitaryrestrictionontheconstraintlanguagesisunavoidable,sinceotherwisethe
arityoftherelationsorthedomainsizewouldbeunbounded. However,forunboundedarity,smallbackdoors
cannot be found efficiently as Lemma 1 would not hold already for the special case of d = 1 unless
FPT = W[2][20]. Similarly,withunboundeddomain,asmallstrongbackdoorcannotbeusedefficiently.
Forinstance,thenaturalencodingoftheW[1]-hardk-cliqueproblemtoCSP[34]onlyhask variables,and
thereforehasasize-k strongbackdoortoanybaseclassthatcontainsthetrivialconstrainswithemptyscopes,
whichisthecaseforanynaturalbaseclass;anFPTalgorithmsolvingsuchinstanceswouldonceagainimply
FPT = W[1].
ThefollowingisabriefsummaryofthealgorithmofLemma1. Wewillgiveamoredetailedsummaryin
Section3.
1. Webeginbyusingthetechniqueofiterativecompression[37]totransformtheproblemintoastructured
subproblemwhichwecallEXTENDED SBD(CSP(Γ1) CSP(Γd)) COMPRESSION(EXT-SBD
⊕···⊕
COMP). In this technique, the idea is to start with a sub-instance and a trivial solution for this sub-
instance and iteratively expand the sub-instances while compressing the solutions till we solve the
problemontheoriginalinstance. Specifically,inEXT-SBD COMPwearegivenadditionalinformation
aboutthedesiredsolutionintheinput: wereceivean“old”strongbackdoorwhichisslightlybigger
than our target size, along with information about how this old backdoor interacts with our target
solution. ThisisformalizedinSubsection3.1.
2. In Subsection 3.2, we consider only solutions for EXT-SBD COMP instances which have a certain
‘inseparabilityproperty’andgiveanFPTalgorithmtotestforthepresenceofsuchsolutions. Tobe
3
moreprecise,hereweonlylookforsolutionsofEXT-SBD COMPwhichleavetheomittedpartofthe
oldstrongbackdoorinasingleconnectedcomponent. Wehandlethiscaseseparatelyatthebeginning
sinceitservesasabasecaseinouralgorithmtosolvegeneralinstances. Interestingly,eventhisbase
caserequirestheextensionofstateoftheartseparatortechniquestoaCSPsetting.
3. Finally,inSubsection3.3weshowhowtohandlegeneralinstancesof EXT-SBD COMP. Thispartof
thealgorithmreliesonanewpatternreplacementtechnique,whichsharescertainsuperficialsimilarities
withprotrusionreplacement[2]butallowsthepreservationofamuchlargersetofstructuralproperties
(such as containment of disconnected forbidden structures and connectivity across the boundary).
Weinterleaveourpatternreplacementprocedurewiththerecentlydevelopedapproachof‘important
separatorsequences’[30]aswellasthealgorithmdesignedintheprevioussubsectionfor‘inseparable’
instancesinordertosolvetheproblemongeneralinstances. Beforeweconcludethesummary,we
would like to point out an interesting feature of our algorithm. At its very core, it is a branching
algorithm;inFPTtimeweidentifyaboundedsetofvariableswhichintersectssomesolutionandthen
branchonthisset. NotethatthisapproachdoesnotalwaysresultinanFPT-algorithmforcomputing
strongbackdoorsets. Infact,dependingonthebaseclassitmightonlyimplyanFPT-approximation
algorithm (see [22]). This is because we need to explore all possible assignments for the chosen
variable. However,wedevelopanotionofforbiddensetsofconstraintswhichallowsustosuccinctly
describe when a particular set is not already a solution. Therefore, when we branch on a supposed
strongbackdoorvariable,wesimplyaddittoapartialsolutionwhichwemaintainandthenwecan
at any point easily check whether the partial solution is already a solution or not. This is a crucial
componentofourFPTalgorithm.
Related Work Williams et al. [40, 41] introduced the notion of backdoors for the runtime analysis of
algorithmsforCSPandSAT,seealso[25]foramorerecentdiscussionofbackdoorsforSAT.Abackdooris
asmallsetofvariableswhoseinstantiationputstheinstanceintoafixedtractableclass. Onedistinguishes
betweenstrongandweakbackdoors,wherefortheformerallinstantiationsleadtoaninstanceinthebaseclass,
andforthelatteratleastoneleadstoasatisfiableinstanceinthebaseclass. Backdoorshavebeenstudiedunder
adifferentnamebyCramaetal.[11]. Thestudyoftheparameterizedcomplexityoffindingsmallbackdoors
wasinitiatedbyNishimuraetal.[33]forSAT,whoconsideredbackdoorsintotheclassesofHornandKrom
CNFformulas. FurtherresultscovertheclassesofrenamableHornformulas[36],q-Hornformulas[21]and
classes of formulas of bounded treewidth [22, 35]. The detection of backdoors for CSP has been studied
forinstancein[1,7]. Gaspersetal.[20]recentlyobtainedresultsonthedetectionofstrongbackdoorsinto
heterogeneousbaseclassesoftheformCSP(Γ ) CSP(Γ )whereforeachinstantiationofthebackdoor
1 d
∪···∪
variables,thereducedinstancebelongsentirelytosomeCSP(Γ )(possiblytodifferentCSP(Γ )’sfordifferent
i i
instantiations). OursettingismoregeneralsinceCSP(Γ ) CSP(Γ ) CSP(Γ ) CSP(Γ ),
1 d 1 d
⊕···⊕ ⊇ ∪···∪
andthesizeofasmalleststrongbackdoorintoCSP(Γ ) CSP(Γ )canbearbitrarilylargerthanthe
1 d
∪···∪
sizeofasmalleststrongbackdoorintoCSP(Γ ) CSP(Γ ).
1 d
⊕···⊕
2 Preliminaries
2.1 ConstraintSatisfaction
Let be an infinite set of variables and a finite set of values. A constraint of arity ρ over is a pair
V D D
(S,R)whereS = (x ,...,x )isasequenceofvariablesfrom andR ρ isaρ-aryrelation. Theset
1 ρ
V ⊆ D
var(C) = x ,...,x iscalledthescopeofC. Avalueassignment(orassignment,forshort)α : X
1 ρ
{ } → D
4
is a mapping defined on a set X of variables. An assignment α : X satisfies a constraint
⊆ V → D
C = ((x ,...,x ),R) if var(C) X and (α(x ),...,α(x )) R. For a set I of constraints we write
1 ρ 1 ρ
(cid:83) ⊆ ∈
var(I) = var(C)andrel(I) = R : (S,R) C,C I .
C∈I { ∈ ∈ }
AfinitesetIofconstraintsissatisfiableifthereexistsanassignmentthatsimultaneouslysatisfiesallthe
constraintsinI. TheConstraintSatisfactionProblem(CSP,forshort)asks,givenafinitesetIofconstraints,
whetherIissatisfiable. TheCountingConstraintSatisfactionProblem(#CSP,forshort)asks,givenafinite
setIofconstraints,todeterminethenumberofassignmentstovar(I)thatsatisfyI. CSPisNP-completeand
#CSPis#P-complete(see,e.g.,[5]).
Letα : X beanassignment. Foraρ-aryconstraintC = (S,R)withS = (x ,...,x )wedenote
1 ρ
→ D
byC theconstraint(S(cid:48),R(cid:48))obtainedfromC asfollows. R(cid:48) isobtainedfromRby(i)deletingalltuples
α
|
(d ,...,d )fromRforwhichthereissome1 i ρsuchthatx X andα(x ) = d ,and(ii)removing
1 ρ i i i
≤ ≤ ∈ (cid:54)
fromallremainingtuplesallcoordinatesd withx X. S(cid:48) isobtainedfromS bydeletingallvariablesx
i i i
∈
withx X. ForasetIofconstraintswedefineI as C : C I .
i α α
∈ | { | ∈ }
Aconstraintlanguage(orlanguage,forshort)Γoverafinitedomain isasetΓofrelations(ofpossibly
D
variousarities)over . ByCSP(Γ)wedenoteCSPrestrictedtoinstancesIwithrel(I) Γ. Aconstraint
D ⊆
languageΓistractableifforeveryfinitesubsetΓ(cid:48) Γ,theproblemCSP(Γ(cid:48))canbesolvedinpolynomial
⊆
time. AconstraintlanguageΓis#-tractableifforeveryfinitesubsetΓ(cid:48) Γ,theproblem#CSP(Γ(cid:48))canbe
⊆
solvedinpolynomialtime.
Inhisseminalpaper[38],SchaefershowedthatforallconstraintlanguagesΓovertheBooleandomain
0,1 , CSP(Γ) is either NP-complete or solvable in polynomial time. In fact, he showed that a Boolean
{ }
constraintlanguageΓistractableifandonlyatleastoneofthefollowingpropertiesholdsforeachrelation
R Γ: (i)(0,...,0) R,(ii)(1,...,1) R,(iii)Risequivalenttoaconjunctionofbinaryclauses,(iv)R
∈ ∈ ∈
is equivalent to a conjunction of Horn clauses, (v) R is equivalent to a conjunction of dual-Horn clauses,
and (vi) R is equivalent to a conjunction of affine formulas; Γ is then called 1-valid, 0-valid, bijunctive,
Horn, dual-Horn, or affine, respectively. A Boolean language that satisfies any of these six properties is
calledaSchaeferlanguage. AconstraintlanguageΓoverdomain isconservativeifΓcontainsallunary
D
relations over . Except for the somewhat trivial 0-valid and 1-valid languages, all Schaefer languages
D
are conservative. Γ is semi-conservative if it contains all unary relations over that are singletons (i.e.,
D
constraintsthatfixthevalueofavariabletosomeelementof ).
D
AconstraintlanguageΓisclosedunderassignmentsifforeveryC = (S,R)suchthatR Γandevery
∈
assignmentα,itholdsthatR(cid:48) ΓwhereC = (S(cid:48),R(cid:48)). ForaconstraintlanguageΓoveradomain we
α
∈ | D
denotebyΓ∗thesmallestconstraintlanguageover thatcontainsΓ 2 andisclosedunderassignments;
D ∪{D }
noticethatΓ∗isuniquelydeterminedbyΓ. Evidently,ifalanguageΓistractable(or#-tractable,respectively)
andsemi-conservative,thensoisΓ∗: first,allconstraintsoftheform(S, 2 )canbedetectedinpolynomial
α
D |
timeandremovedfromtheinstancewithoutchangingthesolution,andtheneachconstraintC(cid:48) = (S(cid:48),R(cid:48))
withR(cid:48) Γ∗ ΓcanbeexpressedintermsoftheconjunctionofaconstraintC = (S,R)withR Γand
∈ \ ∈
unaryconstraintsovervariablesinvar(C) var(C(cid:48)).
\
As mentioned in the introduction, the union of two tractable constraint languages is in general not
tractable. Take for instance the conservative languages Γ = 0,1 3 (1,1,1) 2{0,1} and Γ =
1 2
{{ } \ { }} ∪
0,1 3 (0,0,0) 2{0,1}. Using the characterization of Schaefer languages in terms of closure
{{ } \ { }} ∪
properties (see, e.g., [23]), it is easy to check that Γ is Horn and has none of the five other Schaefer
1
properties;similarly,Γ isdual-HornandhasnoneofthefiveotherSchaeferproperties. Hence,iffollowsby
2
Schaefer’sTheoremthatCSP(Γ )andCSP(Γ )aretractable,butCSP(Γ Γ )isNP-complete. Onecan
1 2 1 2
∪
findsimilarexamplesforotherpairsofSchaeferlanguages.
5
2.2 ParameterizedComplexity
A parameterized problem is a problem whose instances are tuples (I,k), where k N is called the
P ∈
parameter. Wesaythataparameterizedproblemisfixedparametertractable(FPTinshort)ifitcanbesolved
byanalgorithmwhichrunsintimef(k) I O(1) forsomecomputablefunctionf;algorithmswithrunning
·| |
timeofthisformarecalledFPTalgorithms. ThenotionsofW[i]-hardness(fori N)arefrequentlyusedto
∈
showthataparameterizedproblemisnotlikelytobeFPT;anFPTalgorithmforaW[i]-hardproblemwould
implythattheExponentialTimeHypothesisfails[8]. Wereferthereadertoothersources[15,16,19]foran
in-depthintroductionintoparameterizedcomplexity.
2.3 Backdoors,IncidenceGraphsandScatteredClasses
LetIbeaninstanceofCSPover andlet beaclassofCSPinstances. AsetB ofvariablesofIiscalleda
D H
strongbackdoorinto ifforeveryassignmentα : B itholdsthatI . Noticethatifwearegiven
α
H → D | ∈ H
astrongbackdoorB ofsizek intoatractable(or#-tractable)class ,thenitispossibletosolveCSP(or
H
#CSP)intime k nO(1). Itisthusnaturaltoaskforwhichtractableclasseswecanfindasmallbackdoor
|D| ·
efficiently.
STRONG BACKDOOR DETECTION INTO (SBD( ))
H H
Setting: Aclass ofCSPinstancesoverafinitedomain .
H D
Instance: ACSPinstanceIover andanon-negativeintegerk.
D
Task: FindastrongbackdoorinIinto ofcardinalityatmostk,ordeterminethatnosuch
H
strongbackdoorexists.
Parameter: k.
We remark that for any finite constraint language Γ, the problem SBD(CSP(Γ)) is fixed parameter
tractableduetoasimplefolklorebranchingalgorithm. Ontheotherhand, SBD(CSP(Γ(cid:48)))isknowntobe
W[2]-hardforawiderangeofinfinitetractableconstraintlanguagesΓ(cid:48) [20]. GivenaCSPinstanceI,we
use (I) = (var(I) I,E)todenotetheincidencegraphofI;specifically,Icontainsanedge x,Y for
B ∪ { }
x var(I),Y Iifandonlyifx var(Y). Wedenotethisgraphby whenIisclearfromthecontext.
∈ ∈ ∈ B
Furthermore,forasetS ofvariablesofI,wedenoteby (I)thegraphobtainedbydeletingfrom (I)the
S
B B
verticescorrespondingtothevariablesinS;wemayalsouse inshortifIisclearfromthecontext. We
S
B
refertoDiestel’sbook[14]forstandardgraphterminology.
TwoCSPinstancesI,I(cid:48) arevariabledisjointifvar(I) var(I(cid:48)) = . Let ,... beclassesofCSP
1 d
∩ ∅ H H
instances. Thenthescatteredclass istheclassofallCSPinstancesIwhichmaybepartitioned
1 d
H ⊕···⊕H
into pairwise variable disjoint sub-instances I ,...I such that I for each i [d]. Notice that this
1 d i i
∈ H ∈
implies that (I) can be partitioned into pairwise disconnected subgraphs (I ),... (I ). If ,...
1 d 1 d
B B B H H
aretractable,then isalsotractable,sinceeachI canbesolvedindependently. Similarly,If
1 d i
H ⊕···⊕H
,... are#-tractable,then isalso#-tractable,sincethenumberofsatisfyingassignments
1 d 1 d
H H H ⊕···⊕H
ineachI canbecomputedindependentlyandthenmultipliedtoobtainthesolution.
i
We conclude this section by showcasing that a strong backdoor to a scattered class can be arbitrarily
smallerthanastrongbackdoortoanyofitscomponentclasses. Consideronceagainthetractablelanguages
Γ = 0,1 3 (1,1,1) 2{0,1} (Horn) andΓ = 0,1 3 (0,0,0) 2{0,1} (dual-Horn). Then
1 2
{{ } \{ }}∪ {{ } \{ }}∪
foranyk NonecanfindI CSP(Γ ) CSP(Γ )suchthatIdoesnothaveastrongbackdoorofsizek to
1 2
∈ ∈ ⊕
eitherofCSP(Γ ),CSP(Γ ).
1 2
6
3 Strong-Backdoors to Scattered Classes
Thissectionisdedicatedtoprovingourmaintechnicallemma,restatedbelow. Wewouldliketopointout
thattheassumptionregardingtheexistenceofthetautologicalbinaryrelation 2 inthelanguagesismade
D
purelyforeaseofdescriptioninthelaterstagesofthealgorithm.
Lemma1. LetΓ ,...Γ befinitelanguagesoverafinitedomain whichareclosedunderpartialassign-
1 d
mentsandcontain 2. ThenSBD(CSP(Γ ) CSP(Γ ))caDnbesolvedintime22O(k) I O(1).
1 d
D ⊕···⊕ | |
Beforeproceedingfurther,weshowhowLemma1isusedtoproveTheorem1.
ProofofTheorem1. LetIbeaninstanceofCSP( ∗). RecallingthedefinitionofΓ∗,weuseLemma1to
D
find a strong backdoor X of size at most k into CSP(Γ∗) CSP(Γ∗) in time 22O(k) I O(1). Since
1 ⊕ ··· ⊕ d | |
CSP(Γ∗) CSP(Γ∗) CSP(Γ ) CSP(Γ ),itfollowsthatanystrongbackdoorintoCSP(Γ )
1 ⊕···⊕ d ⊇ 1 ⊕···⊕ d 1 ⊕
CSP(Γ ) is also a strong backdoor into CSP(Γ∗) CSP(Γ∗). We branch over all the at most
···⊕ d 1 ⊕···⊕ d
k assignmentsα : X ,andforeachsuchαwecansolvetheinstanceI inpolynomialtimesince
α
|D| → D |
CSP(Γ∗) CSP(Γ∗)istractable.
1 ⊕···⊕ d
Forthesecondcase,letIbeaninstanceof#CSP( ∗). Asabove,wealsouseLemma1tocomputea
D
strongbackdoorX intoCSP(Γ∗) CSP(Γ∗)ofsizeatmostk. Wethenbranchoverallatmost k
1 ⊕···⊕ d |D|
assignmentsα : X ,andforeachsuchαwecansolvethe#CSPinstanceI inpolynomialtimesince
α
→ D |
CSP(Γ∗) CSP(Γ∗)is#-tractable;letcost(α)denotethenumberofsatisfyingassignmentsofI for
1 ⊕···⊕ (cid:80)d |α
eachα. Wethenoutput cost(α).
α:X→D
WebeginourpathtowardsaproofofLemma1bystatingthefollowingassumptionontheinputinstance,
whichcanbeguaranteedbysimplepreprocessing. LetρbethemaximumarityofanyrelationinΓ ,...,Γ .
1 d
Observation1. Anyinstance(I(cid:48),k)of SBD(CSP(Γ ) CSP(Γ ))eithercontainsonlyconstraints
1 d
⊕···⊕
ofarityatmostρ+k,orcanbecorrectlyrejected.
Proof. AssumethatI(cid:48)containsaconstraintC = (S,R)ofarityρ(cid:48) > ρ+k. ThenforeverysetX ofatmostk
variables,thereexistsanassignmentα : X suchthatC hasarityρ(cid:48) > ρ,andhenceC CSP(Γ )
α α 1
→ D | | (cid:54)∈ ⊕
CSP(Γd). Henceanysuch(I(cid:48),k)isclearlya NO-instanceof SBD(CSP(Γ1) CSP(Γd)).
···⊕ ⊕···⊕
Organization of the rest of the section. The rest of this section is structured into three subsections. In
Subsection3.1,weuseiterativecompressiontotransformtheSBD problemtargetedbyLemma1intoits
compressed version EXT-SBD COMP. Subsection 3.2 develops an algorithm which correctly solves any
instanceof EXT-SBD COMPwhichhasacertaininseparabilityproperty. Finally,inSubsection3.3wegivea
generalalgorithmforEXT-SBD COMPwhichusesthealgorithmdevelopedinSubsection3.2asasubroutine.
3.1 Iterativecompression
Wefirstdescribeawaytoreducetheinputinstanceof SBD(CSP(Γ ) CSP(Γ ))tomultiple(buta
1 d
⊕···⊕
bounded number of)structured instances, such that solving these instances will leadto a solution for the
inputinstance. Todothis,weusethetechniqueofiterativecompression[37]. Givenaninstance(I,k)of
SBD(CSP(Γ ) CSP(Γ ))whereI = C ,...,C ,fori [m]wedefineC = C ,...,C . We
1 d 1 m i 1 i
⊕···⊕ { } ∈ { }
iteratethroughtheinstances(C ,k)startingfromi = 1,andforeachi-thinstanceweuseaknownsolution
i
7
X ofsizeatmostk+ρtotrytofindasolutionXˆ ofsizeatmostk. Thisproblem,usuallyreferredtoasthe
i i
compressionproblem,isthefollowing.
SBD(CSP(Γ1) CSP(Γd)) COMPRESSION
⊕···⊕
Setting: LanguagesΓ ,...,Γ ofmaximumarityρoveradomain .
1 d
D
Instance: ACSPinstanceI,anon-negativeintegerk andastrongbackdoorsetX var(I)
⊆
intoCSP(Γ ) CSP(Γ )ofsizeatmostk+ρ.
1 d
⊕···⊕
Task: FindastrongbackdoorinIintoCSP(Γ ) CSP(Γ )ofsizeatmostk,orcorrectly
1 d
⊕···⊕
determinethatnosuchsetexists.
Parameter: k.
WhenΓ ,...,Γ areclearfromthecontext,weabbreviateSBD(CSP(Γ ) CSP(Γ ))as SBD
1 d 1 d
⊕···⊕
and SBD(CSP(Γ1) CSP(Γd)) COMPRESSION as SBD COMP. Wereducethe SBD problemtom
⊕···⊕
instancesoftheSBD COMPproblemasfollows. LetI(cid:48) beaninstanceof SBD.Thesetvar(C1)isclearlya
strongbackdoorofsizeatmostρfortheinstanceI1 = (C1,k, )of SBD COMP. Weconstructandsolve
∅
asequenceof SBD COMP instancesI2,...Im bylettingIi = (Ci,k,Xi−1 var(Ci)),whereXi−1 isthe
∪
solution to I . If some such I is found to have no solution, then we can correctly reject for I(cid:48), since
i−1 i
C I(cid:48). On the other hand, if a solution X is obtained for I , then X is also a solution for I(cid:48). Since
i m m m
⊆
therearemsuchiterations,thetotaltimetakenisboundedbymtimesthetimerequiredtosolvetheSBD
COMPproblem.
Moving from the compression problem to the extension version. We now show how to convert an
instanceofthe SBD COMPproblemintoaboundednumberofinstancesofthesameproblemwherewemay
additionallyassumethesolutionwearelookingforextendspartofthegivenstrongbackdoor. Formally,an
instance of the EXTENDED SBD COMP problem is a tuple (I,k,S,W) where I is a CSP instance, k is a
non-negativeintegerandW S isastrongbackdoorsetofsizeatmostk+ρintoCSP(Γ ) CSP(Γ ).
1 d
∪ ⊕···⊕
TheobjectivehereistocomputeastrongbackdoorintoCSP(Γ ) CSP(Γ )ofsizeatmostk which
1 d
⊕···⊕
containsS andisdisjointfromW.
EXTENDED SBD COMPRESSION(EXT-SBD COMP)
Setting: LanguagesΓ ,...,Γ ofmaximumarityρoveradomain .
1 d
D
Instance: ACSPinstanceI,anon-negativeintegerk anddisjointvariablesetsS andW such
thatW S isastrongbackdoorsetintoCSP(Γ ) CSP(Γ )ofsizeatmostk+ρ.
1 d
∪ ⊕···⊕
Task: FindastrongbackdoorinIintoCSP(Γ ) CSP(Γ )ofsizeatmostkthatextends
1 d
⊕···⊕
S andisdisjointfromW,ordeterminethatnosuchstrongbackdoorsetexists.
Parameter: k.
WenowreduceSBD COMPto(cid:0)|X|(cid:1)-manyinstancesof EXT-SBD COMP,asfollows. LetI(cid:48) = (I,k,X)
≤k
beaninstanceof SBD COMP. Weconstruct(cid:0)|X|(cid:1)-manyinstancesof EXT-SBD COMPasfollows. Forevery
≤k
S (cid:0)X(cid:1),weconstructtheinstanceI(cid:48) = (I,k,S,X S). Clearly,theoriginalinstanceI(cid:48) isa YESinstance
∈ ≤k S \
of SBD COMP if and only if for some S (cid:0)X(cid:1), the instance I(cid:48) is a YES instance of EXT-SBD COMP.
∈ ≤k S
Therefore,thetimetosolvetheinstanceI(cid:48) isboundedby(cid:0)|X|(cid:1) 2k+ρ timesthetimerequiredtosolvean
≤k ≤
8
w1 w2 w3 s1 s2 s3
w s s s w
w 2 1 2 3 3
1
Figure1: Anillustrationofseparatingandnon-separatingsolutions. Inbothcases,S = s ,s ,s isthe
1 2 3
{ }
hypotheticalsolutionunderconsiderationwhile w ,w ,w istheoldsolution. Inthefirstfigure,S isa
1 2 3
{ }
non-separatingsolutionwhileinthesecond,itisaseparatingsolution.
instanceofEXT-SBD COMP. Intherestofthepaper,wegiveanFPTalgorithmtosolveEXT-SBD COMP,
whichfollowingourdiscussionaboveimpliesLemma1.
Lemma1.1. EXT-SBD COMPcanbesolvedintime22O(k) I O(1).
| |
WefirstfocusonsolvingaspecialcaseofEXT-SBD COMP,andthenshowhowthishelpstosolvethe
probleminitsfullgenerality.
3.2 Solvingnon-separatinginstances
In this subsection, we restrict our attention to input instances with a certain promise on the structure of
a solution. We refer to these special instances as non-separating instances. These instances are formally
definedasfollows.
Definition 1. Let (I,k,S,W) be an instance of EXT-SBD COMP and let Z S be a solution for this
⊇
instance. WecallZ aseparatingsolution(seeFigure1)forthisinstanceifW isnotcontainedinasingle
connectedcomponentof andanon-separatingsolutionotherwise. Aninstanceiscalledaseparating
Z
B
instanceifitonlyhasseparatingsolutionsanditiscalledanon-separatinginstanceotherwise.
Havingformallydefinednon-separatinginstances,wenowgiveanoverviewofthealgorithmwedesign
to solve such instances. We begin by developing the notion of a forbidden set of constraints. The main
motivation behind the introduction of this object is that it provides us with a succinct certificate that a
particularsetisnotastrongbackdooroftherequiredkind,immediatelygivingusasmallstructurewhich
9
wemustexclude. However,theexclusioninthiscontextcanoccurnotjustbyinstantiatingavariableinthe
scopeofoneoftheseconstraintsinthesolutionbutalsoduetothebackdoordisconnectingtheseconstraints.
Thisissignificantlydifferentfromstandardgraphproblemswhereoncewehaveasmallviolatingstructure,a
straightforwardbranchingalgorithmcanbeusedtoeliminatethisstructure. However,inourcase,evenif
wehaveasmallviolatingstructure, itisnotatallclearhowsuchastructurecanbeutilized. Forthis, we
firstsetupappropriateseparatormachineryforCSPinstances. Wethenarguethatforanyforbiddensetof
constraints,ifavariableinthescopeoftheseconstraintsisnotinthesolution,thenoneoftheseconstraints
mustinfactbeseparatedfromtheoldstrongbackdoorsetbythehypotheticalsolution. Followingthis,we
arguethatthenotionofimportantseparatorsintroducedbyMarx[31]canbeusedtoessentiallynarrowdown
thesearchspaceofseparatorswherewemustsearchforasolutionvariable. Finally,wecanuseabranching
algorithminthissignificantlyprunedsearchspaceofseparatorsinordertocomputeasolution(ifthereexists
one). WereiteratethatthenotionofforbiddensetsiscriticalinobtaininganFPTalgorithmasopposedtoan
FPT-approximationalgorithm. Nowthatwehavegivenaslightlymoredetailedoverviewofthissubsection,
weproceedtodescribeouralgorithmforsolvingnon-separatinginstances. Webeginwiththedefinitionof
forbiddenconstraintsandthensetuptheseparatormachineryrequiredinthisaswellasthenextsubsection.
ForbiddenConstraints. Let(I,k,S,W)beaninstanceof EXT-SBD COMP.
Definition2. LetS var(I),letC = C ,...,C beasetofatmostdconstraintsandJ beasubsetof
1 (cid:96)
⊆ { }
[d]. WesaythatCisJ-forbiddenwithrespecttoS ifthereisanassignmentτ : S suchthatforevery
→ D
i J thereisat [(cid:96)]suchthatC [τ] / Γ . IfJ = [d],thenwesimplysaythatCisforbiddenwithrespect
t i
∈ ∈ ∈
to S. Furthermore, we call τ an assignment certifying that C is J-forbidden (forbidden if J = [d]) with
respecttoS.
Thefollowingobservationisaconsequenceofthelanguagesbeingclosedunderpartialassignments.
Observation2. IfCisasetofconstraintsforbiddenwithrespecttoavariablesetS,thenCisalsoforbidden
withrespecttothesetS var(C). Conversely,ifCisforbiddenwithrespecttoS andS(cid:48) isasetofvariables
∩
disjointfromvar(C),thenCisalsoforbiddenwithrespecttoS S(cid:48) andwithrespecttoS(cid:48).
∪
Theintuitionbehindthedefinitionofforbiddensetsisthatitallowsustohavesuccinctcertificatesfor
non-solutions. Thisintuitionisformalizedinthefollowinglemma.
Lemma2. GivenaCSPinstanceI,asetX var(I)isastrongbackdoorsetintoCSP(Γ ) CSP(Γ )
1 d
⊆ ⊕···⊕
ifandonlyifthereisnoconnectedcomponentof containingasetofconstraintsforbiddenwithrespect
X
B
toX.
Lemma2.1. GivenaCSPinstanceIandasetS ofvariables, wecancheckintime ( |S| I O(1))if
O |D| ·| |
thereisasetofconstraintsforbiddenwithrespecttoS.
Proof. Clearly,itissufficienttorunoverallatmostd-sizedsetsofconstraintsandallassignmentstothe
variables in S and examine the reduced constraints if they belong to each of the languages Γ ,...,Γ .
1 d
Sincetheselanguagesarefinite,thefinalcheckcanbedoneintime (1). Thiscompletestheproofofthe
O
lemma.
Lemma2.2. LetIbeaCSPinstanceandletCbeasetofconstraintscontainedinacomponentofIand
forbidden(withrespecttosomeset). LetZ beastrongbackdoorsettoCSP(Γ ) CSP(Γ )forthis
1 d
⊕···⊕
instance. Then,eitherZ disconnectsCorZ var(C) = .
∩ (cid:54) ∅
10