Acyclic orders, partition schemes and CSPs : Unified hardness proofs and improved algorithms

Contents lists available atScienceDirect

Artificial

Intelligence

www.elsevier.com/locate/artint

Acyclic

orders,

partition

schemes

and

CSPs:

Unified

hardness

proofs

and

improved

algorithms

✩

Peter Jonsson,

Victor Lagerkvist

∗

,

George Osipov

DepartmentofComputerandInformationScience,LinköpingsUniversitet,CampusValla,Sweden

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received29June2020

Receivedinrevisedform25January2021 Accepted12April2021

Availableonline15April2021 Keywords:

Constraintsatisfactionproblems Infinitedomains

Partitionschemes Lowerbounds

Many computational problems arising in, for instance, artificial intelligence can be realized as infinite-domain constraint satisfaction problems (CSPs) based on partitionschemes: a set of pairwise disjoint binary relations (containing the equality relation) whose union spans the underlying domain and which is closed under converse. We first consider partition schemes that contain an acyclic order and where the constraint language contains all unions of the basic relations; such CSPs are frequently occurring in e.g. temporal and spatial reasoning. We identify properties of such orders which, when combined, are sufficient to establish NP-hardness of the CSP and strong lower bounds under the exponential-time hypothesis, even for degree-bounded problems. This result explains, in a uniform way, many existing hardness results from the literature, and shows that it is impossible to obtain subexponential time algorithms unless the exponential-time hypothesis fails. However, some of these problems (including several important temporal problems), despite likely not being solvable in subexponential time, admit non-trivial improved exponential-time algorithm, and we present a novel improved algorithm for RCC-8 and related formalisms.

©2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

In thisarticlewestudy the complexity ofinfinite-domainconstraintsatisfactionproblems over partitionschemes. Inthis framework onecanformulate manynaturally occurringreasoningproblemsinartificialintelligencesuch asAllen’sinterval algebra and theregionconnectioncalculus.Weidentifysharedpropertiesamongtheseproblems,basedontheexistence of acyclicorders,andusethesepropertiestoprovideageneralNP-hardnessresultandprovestronglower boundsunderthe exponential-timehypothesis.Importantly,tothebestofourknowledge,thisisthefirstlowerboundunderthe exponential-time hypothesisforproblemsofthisform.Motivatedby theselowerboundswealsoturntothe problemofconstructing improvedalgorithmsforCSPsoverpartitionschemes,withaparticularfocusontheregionconnectioncalculus.

1.1. Background

The constraintsatisfactionproblem over a constraintlanguage



(CSP

()

) is thedecision problemofverifying whether a set of constraints based on the relations in



admits a satisfying assignment. For finite domains the complexity of

✩ _{Partsofthisarticleappearedintheproceedingsofthe43rdInternationalSymposiumonMathematicalFoundationsofComputerScience(MFCS-2018).}

*

Correspondingauthor.

E-mailaddresses:peter.jonsson@liu.se(P. Jonsson),victor.lagerkvist@liu.se(V. Lagerkvist),george.osipov@liu.se(G. Osipov). https://doi.org/10.1016/j.artint.2021.103505

0004-3702/©2021TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

CSP

()

iswell understood dueto therecent dichotomy theoremseparating tractable fromNP-complete problems [9,42], but for infinite domains the situation differs markedly. This class of problems includes both undecidable problems and NP-intermediateproblems,andit isthereforecommontoimpose additionalassumptions ontheallowed constraints.The predominant method has been to fix a constraint language



, usually satisfying certain model-theoretic properties, and analysethecomplexity ofCSPsoverfirst-orderreductsof



.Traditionally,thishasalsobeenthecaseforCSPsarisingfrom artificialintelligence,e.g.temporalandspatialreasoningproblems,albeitusuallywithweakerclosureconditions.

Motivatedbyproblemsofthisform,westudythecomplexityofinfinite-domainCSPsoverpartitionschemes.Apartition scheme [29] isasetofpairwisedisjointbinary relations

B

overadomain D suchthat



_R∈BR

=

D2,theequalityrelation is a member of

B

, andthe converse of R is in

B

for every R

∈

B

. Due to their capability of modellingmany different kinds ofqualitativereasoning problems,partition schemesare thede facto standardforCSPsin theartificialintelligence community [13].Givenapartitionscheme,thepredominantwayofformingnewrelationsistoallowunionsoftherelations in

B

, and we let

B

∨= denote this set. Equivalently, each relation in

B

∨= can be defined as a disjunction of the form B1

(x,

y)

∨

B2

(x,

y)

∨ · · · ∨

Bk

(x,

y)forsome

{

B1

,

. . . ,

Bk

}

⊆

B

,andtheset

B

∨= thuscontains allrelationsdefinableinthis way.

Famous AIexamples offormalismsbasedonpartition schemesincludeAllen’sintervalalgebra,the regionconnection cal-culus (RCC-8), and therectanglealgebra.For moreexamples, see e.g. the survey by Dylla et al. [14]. CSP

(B

∨=

)

problems havebeenproventobeNP-hardformanychoicesof

B

.Theproofshaveutilisedvariousreductionsfromvariousproblems, butthere has not beena clearexplanation why themajority ofthem are NP-hard.Thus, doesthere exist one reduction applicable to every partition scheme?Or does oneneed separate proofsfor(e.g.) RCC-8 andAllen’sinterval algebra, and thenperformanexhaustivecaseanalysisonallpossiblesetofbaserelations

B

?

Thus,while quiteabitisknownaboutspecificpartition schemes,itissafetosaythatwe lackamoregeneral under-standing ofwhy theseproblemsare NP-hard.Whenturningto questionsofmorefine-grainedcomplexitythesituationis evenmoredire.Forexample,canCSPsoverpartitionschemesbe solvedroughlyasfastasBooleansatisfiabilityproblems, i.e.,inO

(

2n

₎

_{time?Oratleastasfastasafinite-domainCSP,i.e.,O}

_(c

n

₎

_forsomec

_≥

_{1?Classifyingpartitionschemes} accord-ingtothishighly“fine-grained”levelofcomplexityisaveryhardopenproblemwhichweshouldnothopetoimmediately resolve,andasafirstsanitycheckitiscommontofirstruleout theexistenceofsubexponential algorithms,i.e.,algorithms witha runningtime of2o(n)_{.Naturally,thiscannot be done unconditionally,anda popularassumption inthiscontextis}

theexponential-timehypothesis (ETH)whichstatesthatthe3-SATproblemcannotbesolvedinsubexponentialtime.Despite thefactthat CSPsoverpartition schemesareamongthemostfrequentlystudiedCSPsinartificialintelligence,itisfairto saythat such ETH-basedlower boundshavelargely beenneglectedby both theartificialintelligencecommunityandthe CSPcommunity,andtothebestofourknowledgethereare no concretelowerboundsundertheETH fortheseproblems. There area fewreasonsforthis. First,significanteffortshavebeenmadeto solvehardreasoning problemswithefficient heuristics [35],which are typically difficultto analyse rigorouslyevenif they work well forcertain real-world instances. Second, existinglower boundsare typicallybasedonsize-preservingreductionsfromSAT-like problemswhereoneneeds theabilitytoexpressdisjunctiveclauses,whichisdifficulttoexpresswithpartitionschemes.Tothebestofourknowledge, theonlyconcretelowerboundsforaCSPoverapartitionschemeistheboundbyJonssonandLagerkvist [25] whichrelates the complexity ofAllen’sinterval algebrato thecomplexity ofthe ChromaticNumberproblem(butnot underthe more

commonassumptionETH).

1.2. Ourresults

Ourfirst step(inSection 3) istonote that themajorityofpractically relevantpartition schemescontain acyclicorders satisfyingcertainproperties,whichwe inthisarticlerefertoasunboundedtotalorders,in-forks,out-forks,andno-forks.For brevity, we typically refer to these conditions as C1

,

C2

,

C3 and C4. We provide several examples fromthe literature of

partition schemescontaining acyclicorders ofthisform, e.g.,Allen’sintervalalgebra,theunit intervalalgebra, andRCC-8. While thereis nogeneralrecipe forprovingthat a partition schemecontains an acyclicorderof theabove form, all the examplesthatwehavetriedshowthatitinpracticeisratherstraightforward.

InSection4weturntotheproblemofprovinglowerboundsforCSPsoverpartitionschemescontaininganacyclicorder satisfyingC1–C4.Throughasophisticatedreductionweprovethatnoproblemofthisformcanbesolvedinsubexponential

time(undertheETH),eveninthestructurallyrestrictedcasewhereavariablemayoccurinatmost3constraints.Thecase whenavariablemayoccurinatmost2constraintsishandledinSection4.3whereweestablishthatallCSPsofthisform aretractableifthetemplateinquestionis

ω

-categorical.

In particular, our results implythat CSPs over partition schemes containing acyclic orders of thisform are NP-hard. Hence, wesucceedbothinfindingauniformhardnessproof,andwithproving lowerboundsundertheETH.Importantly, ourlower boundsarethefirst ETH-basedlowerboundsforCSPsofthisform. Itmightalsobeinteresting toobservethat wedonotneedanystrongmodel-theoreticproperties,e.g.

ω

-categoricity,whichisotherwisecommonforinfinite-domain CSPs. The reduction establishingNP-hardness isalso interesting to compareto theprocedure by RenzandLi [36] which takesa partition schemeasinputandtriestoprove NP-hardness.Oneimportantdistinction isthat ourresultprovidesa concretesource ofNP-hardnesswhilethealgorithminRenzandLigivesnosuch insight.Moreover,thisprocedureisnot complete,andisduetocomputationalconstraintsnotapplicabletoe.g.therectanglealgebra,whileitisastraightforward

tasktoprovethatthisalgebrafallswithinthescopeofourresult.Hence,ourstudyoffersamoretheoreticalexplanationof whysomanynaturallyoccurringCSPsoverpartitionschemesarecomputationallyhard.

Onewayofinterpretingtheseresultsisthat CSP

(

B

∨=

)

,when

B

isapartition schemecontainingan acyclicorder sat-isfyingtheaforementioned properties,isfarfrombeingpolynomial-timesolvable:thereisaconstant c

>

1 suchthat the problemcannot besolved in O

(c

n

₎

_{time.An immediateconsequenceoflowerboundsofthisformisthatwe can} imme-diatelyruleoutcertainkindsofalgorithmsforCSP

(

B

∨=

)

,e.g.algorithmsbasedongraph-decompositionandk-consistency, whichtypicallyruninsubexponentialorpolynomialtime.Itisofcoursetemptingtostrengthenourlowerboundeven fur-thersincethecurrentbestknownalgorithmforCSP

(B

∨=

)

foranarbitrarypartitionscheme

B

runsin2O(n2₎

time,ifCSP

(B)

ispolynomial-timesolvable [25,39].Improvementsto2O(n log n)_{arepossibleforcertaintemporalCSPs andforAllen’sinterval}

algebra [25],sothereisreasonforbeingoptimistic.WeattackthisquestioninSection5wherewebeginbypresentingan 2O(n)timealgorithmforthedegree-boundedcase,andthenpresentanovel2O(n log n)timealgorithm forRCC-8.Thisisthe firstexampleofanon-trivialspatialpartitionschemewhoseCSPcanbesolvedin2o(n2₎

time.Classifyingpartitionschemes admittingimprovedalgorithmsofthisformisaninterestingopenquestion,whichwediscussingreaterdetailinSection6, amongotherunresolvedquestions.

2. Preliminaries

In this section we introduce the necessary prerequisites concerning constraint satisfaction problem and partition schemes.WebeginbydefiningtheCSPproblemwhenitisparameterizedbyasetofrelations.

Definition1.Let



beasetoffinitaryrelationsoversomesetD ofvalues.Theconstraintsatisfactionproblem over



(CSP

()

) isdefinedasfollows:

Instance: Aset V ofvariablesandasetC ofconstraints oftheformR(v1

,

. . . ,

vk

)

,wherek isthearityof R,v1

,

. . . ,

vk

∈

V andR

∈ 

.

Question: Isthereafunction f

:

V

→

D suchthat

(

f

(v

1

),

. . . ,

f

(v

k

))

∈

R forevery R(v1

,

. . . ,

vk

)

∈

C ?

The set



iscalleda constraintlanguage,and thefunction f is sometimescalleda satisfyingassignment, ora solution. Importantly,thedomainD isallowedtobeinfinite,andCSPsoverconstraintlanguagesoverinfinitedomainsaretypically calledinfinite-domainCSPs, tocontrastthem withfinite-domain CSPs.Givenan instance I of CSP

()

we write

||

I

||

forthe numberofbitsrequiredtorepresentI.Wewilloccasionallyencounterbounded-degree CSPinstances.Let

(V

,

C

)

denotean instance ofCSP

()

. Ifa variable x occurs in B (distinct)constraints in C ,then we say that thedegree of x is B. Welet CSP

()

-B denote theCSP

()

problemwhereeachvariableintheinputisrestrictedtohavedegreeatmost B.Notethatif

(V

,

C)isaCSP

()

-B instance,then

|

C

|

≤

B

· |

V

|

,implyingthat thenumberofconstraintsislinearlyboundedwithrespect tothenumberofvariables.

Wearenowreadytointroducepartitionschemes [29].Let

B = {

B1

,

. . . ,

Bm

}

beaconstraintlanguageconsistingofbinary relations over a domain D. We saythat

B

is jointlyexhaustive if



B =

D2 andthat

B

ispairwisedisjoint if Bi

∩

Bj

= ∅

wheneveri

=

j.Wesaythat

B

isapartitionscheme if(1)

B

isjointly exhaustiveandpairwisedisjoint,(2)eq_D

= {(

x,x)

|

x

∈

D

}

∈

B

,and(3)forevery Bi

∈

B

,theconverserelation B_i (i.e. B_i

= {(

y,x)

|

(x,

y)

∈

Bi

}

)isin

B

.Wedefine

B

∨=tobethe setofall unionsofrelationsfrom

B

.Notethat eachsuch relationisstillbinary,i.e.,ofarity2.Equivalently,each relation in

B

∨= canbeviewedasadisjunctionB1

(x,

y)

∨

B2

(x,

y)

∨ · · · ∨

Bk

(x,

y)forsome

{

B1

,

. . . ,

Bk

}

⊆

B

.Wesometimesabuse notationandwrite

(B

1

,

. . . ,

Bk

)

todenotetherelation B1

∪ · · · ∪

Bk.Theset

B

∨=andtheproblemCSP

()

where



⊆

B

∨= aretypicalobjectsthatarestudiedintheartificialintelligenceliterature.Forexample,ithasbeencommontousearelation algebra

A

asa startingpoint andthen defineanetworksatisfactionproblem over

A

,whichinournotation isnothingelse thantheCSPoverasetofbinaryrelations.NotethatifCSP

(

B)

ispolynomial-timesolvable,thenCSP

(

B

∨=

)

isamemberof NP.

Example1.Allen’sintervalalgebra [2] isawell-known formalismfortemporalreasoningwhereone considersrelations be-tween intervalsoftheform I

= [

I−

,

I+

]

,where I−

,

I+

∈ R

is thestart andendpoint, respectively.In Allen’salgebraone canforinstancedescribethatoneintervalbeginsbeforeanotherinterval,andoneexpresssuchrelationsintermsofa par-titionschemeconsistingof13basicrelations(seeTable1),andthenformmorecomplicatedrelationsbytakingtheunion ofthe basicrelations.Ifwelet

A

denotethe setof13basicrelationsin Allen’salgebra, thenCSP

(

A

∨=

)

isan alternative formulationofthenetworkconsistencyproblemoverAllen’salgebra.NotethatCSP

(A

∨=

)

isaninfinite-domainCSPsince theunderlyingdomainofrealintervalsisintrinsicallyinfinite.

An extension of the interval algebra is the so-called rectanglealgebra [19,32]. Here, one considers relations between rectanglesintheplanebyextendingthebasicrelationsintheintervalalgebratotheprojectionsofarectangleontothe x-andy-axis,respectively.Inotherwords,givenr,s

∈

A

andtworectanglesrepresentedbytheintervalsIx

,

Iy, Jx

,

Jywemay definetherelationr

⊕

s intherectanglealgebraholdingifIx

(r)

Jx andIy

(s)

Jy.

Example2.RCC-8 [34] isaformalism forqualitativespatial reasoningwherethe basicobjects(referredto asregions)are non-empty regularclosedsubsetsofatopologicalspace. Theregions donothavetobeinternally connected,thatis,they

Table 1

ThethirteenbasicrelationsinAllen’sintervalalgebra.The endpointrelations I−<I+ and J−<J+ thatarevalidfor allrelationshavebeenomitted.

Basic relation Example Endpoints x precedes y p xxx I+<J− y preceded by x p−1 yyy x meets y m xxxx I+=J− y met-by x m−1 yyyy x overlaps y o xxxx I−<J−<I+, y overl.-by x o−1 yyyy I+<J+ x during y d xxx I−>J−, y includes x d−1 _{yyyyyyy I}+_<J+ x starts y s xxx I−=J−, y started by x s−1 yyyyyyy I+<J+ x finishes y f xxx I+=J+, y finished by x f−1 yyyyyyy I−>J− x equals y ≡ xxxx I−=J−, yyyy I+=J+ X Y EQ(X,Y). X Y PO(X,Y). X Y NTPP(X,Y). Y X NTTP−1(X,Y). X Y EC(X,Y). X Y DC(X,Y). X Y TPP(X,Y). Y X TPP−1(X,Y).

Fig. 1. Illustration of the basic relations of RCC-8 with two-dimensional disks.

may consist ofdifferent disconnected pieces. RCC-8 contains eight basic relations: EQ (equal), PO (partial overlap), DC (disconnected), EC (externallyconnected), NTPP (non-tangentialproperpart),itsconverseNTPP−1, TPP (tangentialproper part) andits converse TPP−1. SeeFig. 1for examples.RCC-5 is a variantof RCC-8 whereone is not able todistinguish regions fromtheir topologicalclosure, i.e.the distinction betweenboundary points andinterior points is ignored. Thus, the disconnectednessrelations DC and EC are replacedby DR

=

DC

∪

EC, thetangential andnon-tangentialproper part relations TPP and NTPP arereplacedbyPP

=

TPP

∪

NTPP,and PP−1_{isdefinedanalogously.}

Last,wedefinetwosatisfiabilityproblemsusefulinthecontextoflowerbounds,whichwewillreturntoinSection4.1. If V isasetofvariablesand f a functionfromV to

{

0

,

1

}

,thenwedefine thefunctionhf ashf

(x)

=

f

(x)

andhf

(

¬

x)

=

1

−

f

(x)

foranyx

∈

V .Fork

≥

1 wedefinethek-satisfiability (k-SAT)problemasfollows.

Instance: Asetofvariables V andasetclausesoftheform

(

1

∨ . . . ∨ 

k

)

,where



i

= ¬

x or



i

=

x forsomex

∈

V . Question: Doesthereexistafunction f

:

V

→ {

0

,

1

}

suchthathf

(

1

)

+ . . . +

hf

(

k

)

≥

1 foreachclause

(

1

∨ . . . ∨ 

k

)

?

Thevariantofk-SATwhereweinadditionrequirethatthefunction f doesnotassignthesamevaluetoeachliteralin anyclauseis knownasthe not-all-equal-k-satisfiability problem (NAE-k-SAT).NAE-3-SAT remainsNP-hardeven ifit is restrictedtoclausescontainingonlypositiveliterals,andweinvite thereadertoverifythatthisrestrictedproblemcanbe formulatedasaBooleanCSPoverthetemplateN

= {

0

,

1

}

3

\ {(

0

,

0

,

0

),

(

1

,

1

,

1

)

}

.

3. Acyclicorders

CSPsbasedonpartitionschemesare oftenusedforqualitativereasoning.Weacknowledgethatitisnotobvioushowto define “qualitative reasoning” rigorously, butthe concept seems tohave an informal meaning that isgenerally accepted. RenzandNebel [39,p.161] write

a b c d1

a b c

d2

Fig. 2. Illustration of in-fork (left) and out-fork (right). Solid arrows denote the≺relation and dotted arrows therelation.

Qualitative reasoningisan approachfor dealingwithcommonsense knowledgewithout usingnumericalcomputation. Instead,one triestorepresentknowledgeusingalimitedvocabulary suchasqualitative relationshipsbetweenentities orqualitativecategoriesofnumericalvalues,...

Abstractionisthedefiningfeatureofqualitativereasoning:qualitative reasoningisaboutdisregardingunnecessaryand uninterestingdetails.Withthisinmind,itisclearthatanimportantkindofqualitativerelationships betweenobjectsare “part-of”relations.Onemayarguethatsuchrelationsareordersthatsatisfycertainadditionalproperties.Atypicalexample of such a relationis the NTPP relation in RCC-8—thiscan be viewed asan archetypical example ofa “part-of” relation. Inspiredbythis, wepresent(inSection3.1)acollectionoffourpropertiesthatcapturesome commonaspectsof“part-of” relations.Manyotherrelations(thatarenotnecessarily“part-of”relations)satisfytheseproperties,too:oneexampleisthe precedesrelation p inAllen’salgebra.Infact,relationsofthiskindappearveryfrequentlyinCSPsforqualitativereasoning and we presenta number ofexamples in Section 3.2. Naturally, whileour conditions on acyclicorders are sufficient to establishhardness,thereexistsCSPsoverothertypesofpartitionschemeswhichareofindependentinterest.Forexample, ifweleave therealmofqualitativereasoning,thenthegraphintervalsandwichproblem [17] canbe phrasedasaCSPover thetworelationsp

∪

p−1and



_R∈A\{p,p−1_}R,neitherofwhichisacyclic.Anotherrelevantexampleisthepartitionscheme

{

eqD

,

neqD

}

over a countably infinite D. While the tractable problemCSP

(

{

eqD

,

neqD

})

might not be terriblyinteresting, theoptimisationvariantofthisproblemwhereoneisinterestedinmaximisingthenumberofsatisfiedconstraints, MAX-CSP

(

{

eqD

,

neqD

})

,canbeviewedasanalternativeformulationofcorrelationclustering [5].

3.1. Conditionsonacyclicorders

Let

≺ ⊆

D2 denoteabinaryrelationandlet



denoteits converse

≺

_.Wesaythat

_≺

_{isanacyclicorder iftheredoes}

not existanyfinitesubset

{

d1

,

. . . ,

dk

}

⊆

D suchthatd1

≺

d2

≺ · · · ≺

dk−1

≺

dk

≺

d1.Acyclicorders areirreflexive(i.e.they

donotcontainanyelementd suchthatd

≺

d)bydefinition.Wesaythat

≺

isastrictpartialorder ifitisirreflexiveandfor arbitraryd,d

,

d

∈

D:d

≺

d andd

≺

d implyd

≺

d (transitivity). Notethatthesetwo propertiesalsoensurethat

≺

is antisymmetric,i.e.ifd

≺

d,thend

≺

d doesnothold.Itiseasytoverifythatstrictpartialordersareacyclicordersbutthat thereexistacyclicordersthatare notstrictpartialorders.Wesaythat

≺

isastricttotalorder if

≺

isastrict partialorder anditisa connex relation,i.e.forarbitrarydistinct d,d

∈

D, eitherd

≺

d ord

≺

d holds.We willnowdefine additional propertiesofacyclicordersparticularlyrelevantinthecontextofCSP

(

B

∨=

)

.Here,weinvitethereadertoviewtherelation



whichholdsbetweentwo elementsifthey arenotcomparable withrespectto

≺

,althoughall thatisneededisthat



satisfiesthefollowingproperties.

Definition2.Let

≺ ⊆

D2 beanacyclicorderand



⊆

D2 arelation.Wedefinethefollowingproperties.

C1. (unboundedtotalorders)foreveryk

∈ N

,thereexistsasubset L

⊆

D suchthat

|

L

|

≥

k and

≺

isastricttotalorderon L,

C2. (in-fork)ifa,b,c

∈

D,a

≺

b

≺

c,anda

≺

c,thenthereexistsd1

∈

D suchthatd1



a,d1



b,andd1

(

≺,

)

c,

C3. (out-fork)ifa,b,c

∈

D anda

≺

b

≺

c,anda

≺

c,thenthereexistsd2

∈

D suchthatd2

(

≺,

)

a,d2



b,andd2



c,and C4. (no-fork)ifa,b,c

∈

D anda

≺

b

≺

c,anda

≺

c,thentheredoesnotexistanyd3

∈

D suchthatd3



a,d3

(

≺,

)

b,and

d3



c.

Relations satisfyingtheseproperties are abundant in theartificial intelligenceliterature, but they haveto the best of ourknowledge notbeenexplicitlyformalized before.The conditionsin-forkandout-fork are illustratedinFig. 2.Givena relation R itistypicallyeasytocheckifitisanacyclicorderthatcontainsaninfinitetotalorder,butcheckingC2–C4 may

needadditionalwork.

Manypartition schemes contain strictpartial orders,andtheseare slightlyeasierto work withthanacyclicrelations, sinceitisalwayspossibletodefinetherelation



inacanonicalway.Givenanorder

≺ ⊆

D2,welet



denoteits incompa-rabilityrelation D2

\



{≺,

,

eqD

}

.If

B

isapartitionschemeand

B

∨=containsanorder

≺

,then



isincludedin

B

∨=,too. Weobtainthefollowingcharacterization.

Theorem3.Let

B

beapartitionschemewithdomainD suchthat

B

∨=containsastrictpartialorder

≺

.If

≺

andtheincomparability relation



satisfyC1–C3thentheysatisfyC4.

a

b

c

d1 d2

Fig. 3. The choice of d1and d2in the unit interval example.

Proof. Assumeto thecontrarythatthere exista,b,c,d3

∈

D suchthat d3



a, d3

(

≺,

)

b,andd3



c.Therelation

≺

isa

strict partialorder soitis transitive.Ifd3

≺

b,thend3

≺

c.Butthend3



c cannot holdsincethe relations

≺

and



are

disjoint.Similarly,ifd3



b,thena

≺

d3andd3



a cannothold.



3.2. Examples

Consider Allen’s algebra and the relation p, i.e. the relation statingthat one interval appears strictly before another interval. Inthiscase,



can bechosen tobe therelationthat holdsifandonlyiftwo distinctintervalshaveatleastone point incommon. The relation pisclearly acyclic (in fact,it isa strict partialorder) andit contains manyinfinitestrict total orders such as T

= {[

0

,

1

],

[

2

,

3

],

[

4

,

5

],

. . .

}

.Pick threeintervals Ij

= [

I−j

,

I+j

]

∈

T , 1

≤

j

≤

3, such that I1

(

p

)I

2

(

p

)I

3.

The transitivityof p implies that I1

(

p

)I

3,too.Forin-fork,we choose I4

= [

I−1

,

I2+

]

sothat I4



I1, I4



I2,and I4

≺

I3.For

out-fork,onemaychoose I5

= [

I−2

,

I+3

]

.Concerningtheno-forkproperty,simplynotethataninterval I6 whichprecedesI2

(orisprecededbyI2)cannotshareapointwithboth I1 andI3.

We continue with a more complex example that is based on acyclic orders instead of strict partial orders. The Unit IntervalAlgebra (UIA) is Allen’sinterval algebra restrictedto intervals of length one. The UIA has importantapplications in,forinstance, bioinformatics [33]. Consider theoverlaps relationo. Thisrelationis irreflexivebutitis nottransitive in general:theunitintervals

[

0

,

1

]

,

[

0

.

9

,

1

.

9

]

,and

[

1

.

8

,

2

.

8

]

areexamplesofthis.Hence,itisnotapartialorder.Weshowthat oisanacyclicorderthatsatisfiesC1–C4.Thefactthatoisanacyclicorderiseasytoverify.Choosetherelation

≺

toequal

oandlet



holdifandonlyiftwointervalsdonothaveacommonpoint,i.e.,



= (

p

,

p−1

)

.

C1. Weclaim thattheorder

(L,

o

)

whereL

= {[

x,x

+

1

]

|

0

<

x

<

1and x

∈ Q}

isan infinitestrict totalorder—thisimplies

property C1.Irreflexivityisobviousso wecontinue withtheconnexity property.We seethat if

[

a−

,

a+

],

[

b−

,

b+

]

are

distinctmembersofL,theneither

−

1

<

a−

−

b−

<

0 or0

<

a−

−

b−

<

1.Inthefirstcase,itfollowsthata−

<

b−and 1

+

a−

>

b−whichimpliesthata+

=

1

+

a−

>

b−

>

a− anda(o

)b.

Theothercaseisanalogous.Toverifythat

(L,

o

)

is transitive,we arbitrarily pickthreedistinct unit intervalsa,b,c

= [

a−

,

a+

],

[

b−

,

b+

],

[

c−

,

c+

]

inL such that a(o

)b(

o

)c.

Weneedtoverifythata−

<

c−andc−

<

a+

<

c+.Thefactthata(o

)b(

o

)c implies

thata−

<

b−

<

c−and,consequently, thata+

<

c+sincea+

=

a−

+

1 andc+

=

c−

+

1.Finally,notethatc−

−

a−

<

1 soc−

− (

a+

−

1

)

<

1 andc−

<

a+. C2

/C

3. We begin by arbitrarily choosing three distinct unit intervals a,b,c

= [

a−

,

a+

],

[

b−

,

b+

],

[

c−

,

c+

]

in D such that

a(o

)b(

o

)c and

a(o

)c.

Letd1

= [

c

+_−b+

2

,

c+−b+

2

+

1

]

.Weseethata(p

)d

1,b(p

)d

1,andc(o

)d

1sod1



a,d1



b,andd1

(

≺,

)

c.

Letd2

= [

b

−_−a−

2

−

1

,

b−−a−

2

]

.Weseethatd2

(

o

)a,

d2

(

p

)b,

andd2

(

p

)c.

Thisimpliesthatd2

(

≺,

)

a,d2



b,andd2



c.The

choiceofd1 andd2 isillustratedinFig.3.

C4. Pickthreeunitintervalsa,b,c

= [

a−

,

a+

],

[

b−

,

b+

],

[

c−

,

c+

]

inD suchthata(o

)b(

o

)c and

a(o

)c.

Assumetothecontrary

that thereisad3

∈

D suchthatd3



a,d3

(

≺,

)

b,andd3



c.Ifd3

(

p

)a,

thend3 cannot satisfyd3

(

≺,

)

b sod3

(

p−1

)a.

Similarlyd3

(

p

)c.

Thiscontradictsthefactthata

≺

c.

Letusconsideranotherexamplewherethedomaincontainsthecloseddisksin

R

2_,therelation

_≺

_{isthestrictsubset}

relation, andwhere



holds betweentwo regions if andonly if neither region included inthe other. For each k

≥

1 it is then clear that there exists a set of k regions where

≺

induces a strict total order, e.g., k disks c1

,

c2

,

. . . ,

ck where c1

≺

c2

. . .

≺

ck. Pickthreedisks d1

,

d2

,

d3

∈

D such that d1

≺

d2

≺

d3.Howto choose suitable disks forverifying in-fork

Fig. 4. The dashed circles show possible choices of disks for in-fork (left) and out-fork (right).

d2 alsocontainsd1 (oriscontainedbyd3).Thisexamplecaneasilybeadapted torelations suchas PP inRCC-5, NTPP in

RCC-8,andtherelationd

⊕

dintherectanglealgebra.

The examplespresented aboveare just asmallselection ofpartition schemesthat satisfy propertiesC1–C4 andmany

additionalexamplescanbefound,forinstance,inthesurveybyDyllaetal. [14].Last,letusremarkthatthereareexamples ofstrictpartialordersthatdonothavein- and/orout-forks.Well-knownexamplesaretheless-thanrelation

<

inthe (1-dimensional)pointalgebraandinthebranchingtime algebra.Interestingly,CSP

(

B

∨=

)

ispolynomial-timesolvableinthese twocasesandwewillcomebacktothisobservationattheendofSection 4.1.

4. LowerboundsforCSP

(

B

∨=

)

WewillnowstudythecomputationalcomplexityofCSP

(B

∨=

)

when

B

∨= containsanacyclicorder

≺

andarelation



thatsatisfyC1–C4.Toavoidlengthyformulationsofthiskindweintroducethefollowingsetoftemplates.

Definition4.Welet

H

be thesetofpartition schemes

B

such that (1)CSP

(

B)

issolvableinpolynomial time,and(2)

B

containsanacyclicorder

≺

andarelation



thatsatisfy C1–C4.

Notethat it issufficientthat the partitionscheme contains asingle acyclicorder withtheseproperties:theother re-lations are not relevant as long as CSP

(B)

is tractable. Examples where the connection between acyclic orders andthe complexity oftheresultingCSPsisquitepronouncedcanbefoundin,forinstance,Grignietal. [18],RenzandNebel [37], Moratzetal. [31],andKrokhinetal. [27].Thus,isCSP

(

B

∨=

)

alwaysNP-hardwhen

B ∈ H

,orcanthereexisttractablecases? IfCSP

(

B

∨=

)

isindeedNP-hard,howfastcanit besolved?Mightthereexist someparticularly“easy”partition scheme

B

whereCSP

(B

∨=

)

issolvablein O

(c

n

₎

_{foravery smallconstantc? Or evenin O}

_(c

n

₎

_{time forevery constant c}

_>

_{1,i.e., in} subexponential time1_{? Naturally,wecannothope tounconditionallyprovethataCSP}

_(B

∨=

₎

_{problemisnotsubexponential,}

andit is insteadcommon to provelower bounds subjectedto the assumptionthat a specific problemis not solvablein subexponentialtime.Forthispurposethe3-SATproblem,i.e.,satisfiabilityofclausesoflengthatmost3,hasturnedoutto beaveryusefulstartingpoint.

Definition5.Theconjecturethat 3-SATisnotsolvableinsubexponentialtimeisknownastheexponential-timehypothesis (ETH) [23].

The general idea behind a non-subexponentiality lower bound subjected to the ETH is then similar to a typical NP-hardness proof:oneneedstoprovideasuitablereductionfrom3-SATtotheprobleminquestion.Thecomplicatingfactor, ofcourse,isthatoneneedsreductionspreservingsubexponentialcomplexity,whichcansometimesbemuchmoredifficult to constructthan ordinarypolynomial-time many-onereductions. Moreinformation abouttheETH andits consequences canbefoundinthesurveybyLokshtanovetal. [30].

Example3.Considertheclassicalgadgetreductionfrom4-SATto3-SATwhichreplacesaclauseoftheform

(x

1

∨

x2

∨

x3

∨

x4

)

with

(x

1

∨

x2

∨

y)

∧ (

x3

∨

x4

∨ ¬

y),where y isafreshvariable.Ifoneisnotcarefulthenonemightbeledtobelievethat

thisreduction preservessubexponentialcomplexity since we foreachclause inthe original instanceonlyintroduces one freshvariable.Butwhatiftheinstancecontainsasuperlinear amountofclauseswithrespecttothenumberofvariables, e.g.,aquadraticnumber?Assumingthat3-SATissolvableinO

(c

n

)

timeforsomec

>

0,wheren isthenumberofvariables, thisreductionwouldthenonlysaythat4-SATissolvablein O

(c

n2

)

time,andinparticularwouldnotimplythat4-SATis subexponentialif3-SATissubexponential.

However,itisknownthatthedegree-boundedk-SATproblem(forsomefixed B

>

0)issolvableinsubexponentialtime ifandonlyifk-SATis solvableinsubexponentialtime, usingthepowerfulidea ofsparsification [23]. Ifonethen reduces

fromadegree-boundedproblemthetotalnumberofclausesislinearwithrespecttothenumberofvariables,meaningthat theabovereductionpreservessubexponentialcomplexity.

Impagliazzoetal. [23] introduceamoregeneraltheoryofreductionspreservingsubexponentialcomplexitybutforour purposes,it issufficientwithpolynomial-timemany-onereductionswhichgivenaninstance withn variablesproducean instancewithO

(n)

variables.

4.1. ETH-basedlowerboundsandNP-hardness

Arbitrarilychoose

B

in

H

.Wehavetwochallengestoovercome:first,isitpossibletofindauniform reductionapplicable toeveryCSP

(

B

∨=

)

problem;second,cansuchareduction,ifitevenexists,beusedtoobtainlowerboundsundertheETH? We will reachan affirmative answer toboth ofthese questionsinthis section,and willsee that itis possibleto obtain lowerboundsevenfordegree-boundedCSP

(B

∨=

)

-B problems,indeed,evenfortheseverelyrestrictedproblemCSP

(B

∨=

)

-3 whereavariablemayoccurinatmost3constraints.However,beforeweturntothedetailsweconsideranexamplewhich showsalargedifferencebetweenCSP

(B

∨=

)

problemsandrelatedCSPs,andhighlighttheinvolveddifficulty.

Example4.Fora partitionscheme

B

over thedomain D, let

B

∨k be the setof all relationsdefinable by disjunctions of length at mostk, where each atom is a constraintover

B

. It is easy to see that

B

∨=

⊆

B

∨k _{for some k, but that the} converseisnotnecessarilytrue.Infact,

B

∨k isingeneralmuchmoreexpressivethan

B

∨= inthecontext ofCSPs. Tosee this,considerthefollowingreductionfrom3-SATtoCSP

(B

∨3

)

.Weonlysketchthedetailssincethey arenotimportantfor thesubsequentresults.

1. AssumethatthedomainD containsatleasttwoelements.

2. Since

B

isapartitionschemeitalwayscontainstheequalityrelationeq_D andtheinequalityrelationneq_D overD. 3. Introducetwofreshvariablesxf andxt andconstrainthemasneqD

(x

f

,

xt

)

.

4. Foreach3-clause,e.g.,

(x

1

∨

x2

∨ ¬

x3

)

introducetheconstrainteqD

(x

1

,

xt

)

∨

eqD

(x

2

,

xt

)

∨

eqD

(x

3

,

xf

)

.

Inotherwordsthisreductionisastandardgadgetreductionfrom3-SATtoCSP

(B

∨3

)

whichreplaceseach3-clauseby the corresponding disjunctionover

B

.Moreover,since itonlyintroduces2freshvariablesintotal,itimmediatelyfollowsthat CSP

(

B

∨3

)

cannotbesolvedinsubexponentialtimewithoutviolatingtheETH(recallExample3).Notethatwedonoteven requireanyadditionalassumptionson

B

:itissufficientthatitisapartitionscheme.

ForCSP

(B

∨=

)

the situationismuch moredifficultsince it (ingeneral) isnot possibleto representdisjunctionsofthe required form. We will soon seethat while it is possibleto obtain a suitable reduction to CSP

(B

∨=

)

when

B ∈ H

, it is significantlymorecomplicatedthanthereductioninExample4.Beforeweproceedwiththeactualreductionwedefinea usefulgadget.

Lemma6.Assumethat

B ∈ H.

ThenthereexistsaninstanceofCSP(B∨=

)

withvariables

{

a,b,c,x1

,

x2

}

whichsatisfiesthefollowing properties:

G1. Forarbitraryelementsda

,

db

,

dc

∈

D suchthatda

≺

db

≺

dcandda

≺

dc,thereexistelementsd1

,

d2

∈

D suchthatthefunction s

:

V

→ {

da

,

db

,

dc

,

d1

,

d2

}

definedbys(a)

=

da,s(b)

=

db,s(c)

=

dc,s(x1

)

=

d1ands(x2

)

=

d2isasolutiontotheinstance

(V

,

C

∪ {

a

≺

b,b

≺

c

})

.

G2. Forarbitraryelementsda

,

db

,

dc

∈

D suchthatdc

≺

db

≺

daanddc

≺

da,thereexistelementsd1

,

d2

∈

D suchthatthefunction s

:

V

→ {

da

,

db

,

dc

,

d1

,

d2

}

definedbys(a)

=

dc,s(b)

=

db,s(c)

=

da,s(x1

)

=

d1ands(x2

)

=

d2isasolutiontotheinstance

(V

,

C

∪ {

c

≺

b,b

≺

a

})

.

G3.

(V

,

C

∪ {

b

≺

a,b

≺

c,a(

≺,

)

c

})

isnotsatisfiable. G4.

(V

,

C

∪ {

a

≺

b,c

≺

b,a(

≺,

)

c

})

isnotsatisfiable.

Proof. DefinethegadgetG(a,b,c,x1

,

x2

)

tobetheCSP

(B

∨=

)

instance

(

{

a,b,c,x1

,

x2

}, {

x1



a,x1



b,x1

(

≺, )

c,x2

(

≺, )

a,x2



b,x2



c

}).

We demonstratethat G satisfies G1-G4.Properties G1 and G2 followimmediatelyfromin-forkandout-fork.Toprove G3

andG4,we needtoshowthat whenevera,b,c istotallyordered ina waydifferentfroma

≺

b

≺

c orc

≺

b

≺

a,then the

gadgetisnotsatisfied.Assumeforinstancethatb

≺

a

≺

c.Then,

{

b

≺

a,a

≺

c,x2

(

≺,

)

a,x2



b,x2



c

}

mustbesatisfiableand

thisviolatesproperty C4.Theremainingthreecasescan beruledout analogously.Weconcludethat G hastheproperties G1–G4.



Informally,thegadgetG(a,b,c,x1

,

x2

)

constrainsb tobebetweena andc.Equippedwiththislemma,wearenowready

Lemma7.Let

B ∈ H.

ThenCSP(B∨=

)

isNP-hardanditisnotsolvableinsubexponentialtime,unlesstheETHisfalse.

Proof. Recallthat NAE-3-SAT restrictedtopositiveliteralsmaybeviewedasCSP

(

{

N

})

whereN

= {

0

,

1

}

3

_\{(

₀

_,

₀

_,

₀

_),

₍

₁

_,

₁

_,

₁

₎

_}

_,

andthat this problemisNP-complete. We willshow that there exists a polynomial-timemany-to-onereduction f from CSP

(

{

N

})

toCSP

(

B

∨=

)

.

Arbitrarilychooseaninstance

(A,

T

)

ofCSP

(

{

N

})

andconstructaninstanceI

=

f

((A,

T

))

ofCSP

(B

∨=

)

asfollows: F1: addthevariableM to I,

F2: foreacha

∈

A,addavariablea andtheconstrainta(

≺,

)

M to I,

F3: foreachtripleN(a,b,c)

∈

T ,introducefivevariablesz,x1

,

x2

,

x3

,

x4andadda(

≺,

)

b,a(

≺,

)

c,b(

≺,

)

c,G(a,M,z,x1

,

x2

)

,andG(b,z,c,x3

,

x4

)

toI,whereG isthegadgetfromLemma6.

Clearly,thereductionabovecanbecarriedoutinpolynomialtime.Weproceedwiththecorrectnessproof.

First,assume that s isasolutionto I.Foreacha

∈

A,eithers(a)

≺

s(M)ors(a)



s(M).Wedefine asolutions

:

A

→

{

0

,

1

}

suchthats

(a)

=

0 ifandonlyifs(a)

≺

M.Wecontinuebyprovingthatssatisfieseach N(a,b,c)

∈

T .Assumetothe contrarythat s

(a)

=

s

(b)

=

s

(c)

=

0,i.e. s(a),s(b),s(c)

≺

s(M).We analyse thegadgets G(a,M,z,x1

,

x2

)

, G(b,z,r,x3

,

x4

)

andthefourorderingsthattheyallow.

1. s

(a)

≺

s(M)

≺

s(z)ands(b)

≺

s(z)

≺

s(c).Weseethats(M)

≺

s(z)

≺

s(c)sos

(c)

=

1 andthisisnotpossible. 2. s

(a)



s(M)



s(z)ands(b)

≺

s(z)

≺

s(r).Thisisnotpossiblesinces(a)

≺

s(M).

3. s

(a)

≺

s(M)

≺

s(z)ands(b)



s(z)



s(r).Weseethats(M)

≺

s(z)

≺

s(b)sos

(b)

=

1 andthisisnotpossible. 4. s

(a)



s(M)



s(z)ands(b)



s(z)



s(r).Thisisnotpossiblesinceweknowthats(a)

≺

s(M).

Thecasewhens

(a)

=

s

(b)

=

s

(c)

=

1 canbe ruledoutsimilarly.Weconcludethatatleastonevariableisassigned0, atleastonevariableisassigned1,andtheconstraintN(a,b,c)issatisfied.

Fortheotherdirection,assumethatthereexistsasolutions

:

A

→ {

0

,

1

}

to

(A,

T

)

.Weshowhowtoconstructasolution totheinstanceI.LetA0

⊆

A bethevariablesthatareassigned0 bysandlet A1

⊆

A thatareassigned1.Let

(L,

≺)

denote

astricttotalorderin

(D,

≺)

thatcontains2

|

A

|

+

2 elements

d1

≺

e1

≺

d2

≺

e2

≺ · · · ≺

e|A|

≺

d_|A|+1

≺

e|A|+1

.

Constructs

:

A

∪ {

M

}

→ {

d1

,

. . . ,

d|A|+1

}

suchthats(a)

≺

s(M)ifa

∈

A0 ands(a)



s(M)ifa

∈

A1.Thefunctions satisfies

allconstraintsintroducedinstepF2.WecontinuebytheconstraintsintroducedinstepF3.Consideranarbitraryconstraint N(a,b,c)

∈

T and thecorresponding constraintsin I:wehaveintroduced fivefreshvariables z,x1

,

x2

,

x3

,

x4 andthe

con-straints:(1)a(

≺,

)

b,(2)a(

≺,

)

c,(3)b(

≺,

)

c,(4)G(a,M,z,x1

,

x2

)

,and(5)G(b,z,c,x3

,

x4

)

.Theconstraints

(

1

)

− (

3

)

are

clearly satisfiedby s.Wewillnowshowhowtochooses(z)inordertosatisfyconstraints

(

3

)

and

(

4

)

.Haveinmindthat, forinstanceinconstraint

(

4

)

,itissufficienttochooses(z)suchthats(a)

≺

s(M)

≺

s(z)ors(a)



s(M)



s(z);suitablevalues alwaysexistforx1 andx2duetoG1 andG2.

Lete₊bethee-elementin

(L,

≺)

thatistheimmediatelargerneighbourtotheelement f

(M)

anddefinee₋analogously. Giventwodistincta,b

∈

A,leteabbe anarbitrarye-element in

(L,

≺)

that liesbetweens(a)ands(b).Thefollowingtable summariseshows(z)shouldbechosen.

s

(a)

s

(b)

s

(c)

s(z) 0 0 1 e₊ 0 1 0 e₊ 0 1 1 ebc 1 0 0 ebc 1 0 1 e₋ 1 1 0 e₋

Weconcludethatthefunctions canbeextendedtoasolutiontoI.



Weneedthelaststeppingstonetoprovethestrongerversionofthelemmaabove(recallthatCSP

(B

∨=

)

-3 isthe degree-boundedproblemwhereavariablemayoccurinatmost3 constraints).

Lemma8.Let

B

beapartitionschemeandassume

(V

,

C)isaninstanceofCSP(

B

∨=

).

If

|

C

|

≤

c

|

V

|

forsomeconstantc,then

(V

,

C

)

canbereducedtoaninstanceofCSP(B∨=

)-3 with

atmost2c

|

V

|

variablesinpolynomialtime.

Proof. First,recallthateq_D

∈

B

since

B

isapartitionscheme.Thenpickavariablex

∈

V occurringinconstraintsc1

,

. . . ,

ck fork

>

3.Introducek freshvariables x1

,

. . . ,

xk togetherwiththeconstraintsx1

(

eqD

)x

2,x2

(

eqD

)x

3,. . . , xk−1

(

eqD

)x

k.Next,

replaceeachoccurrenceofx inci bythecorrespondingvariablexi.Clearly,thedegreeofeachxivariableisatmost3,and theequalityconstraintsenforcethatx1

,

. . . ,

xk arealwaysassignedthesamevalueinanysatisfyingassignment.Moreover, eachconstraintcontainstwovariables,sothetotalnumberofvariablesintroducedbythisreductionisboundedfromabove by2

|

C

|

≤

2c

|

V

|

.



Withthislemmaathand,wearenowreadytoprovethemainresultbycarefullyanalysingthereductioninLemma7.

Theorem9.Let

B ∈ H

.ThenCSP(

B

∨=

)-3 is

NP-hardanditisnotsolvableinsubexponentialtime,unlesstheETHisfalse.

Proof. Asbefore, let N

= {

0

,

1

}

3

\ {(

0

,

0

,

0

),

(

1

,

1

,

1

)

}

. Thereexists a constant B

≥

1 suchthat CSP

(

{

N

})

-B is NP-complete andsolvableinsubexponentialtimeifandonlyiftheETHisfalse [26].

Take an arbitrary instance

(A,

T

)

of CSP

(

{

N

})

-B with

|

A

|

=

n (note that

|

T

|

≤

Bn) and apply the reduction f from Lemma 7to obtain an instance

(V

,

C)

=

f

((A,

T

))

ofCSP

(B

∨=

)

.Observe that

|

C

|

= |

A

|

+

5

|

T

|

≤ (

1

+

5B

)n,

since we in-troduce a constrainta(

≺,

)

M for each a

∈

A and five constraints foreach N

(a,

b,c)

∈

T . The term 1

+

5B is constant. Hence,combining f withthereductionfromLemma7yieldsaninstanceofCSP

(

B

∨=

)

-3 with O(n)variablesinpolynomial

time.



Naturally,Theorem9alsoimpliesthatCSP

(B

∨=

)

-3 (and,hence,CSP

(B

∨=

)

)isNP-complete.Forstrictpartialorderswe maycombineTheorem9withtheobservationinTheorem3toobtainthefollowingcorollary.

Corollary10.Let

B

beapartitionschemewithdomainD suchthat

B

∨=containsastrictpartialorder

≺

.Assume

≺

togetherwith theincomparabilityrelation



satisfyC1–C3.ThenCSP(B∨=

)-3

isNP-hardandnotsolvableinsubexponentialtime,unlesstheETHis false.

In summary,wemayrule out subexponentialtime algorithmsforCSP

(B

∨=

)

-3forpartition schemes

B ∈ H

.However, the best general algorithm forCSP

(B

∨=

)

runsin O

(

2O(n2)

)

time (if CSP

(B)

istractable) [25,39]. Hence, there is a large discrepancybetweenthe upperandlower boundforthisproblem, suggestingthat (atleast) oneof theseboundscanbe strengthened.WereturntothisquestioninSection5.

4.2. Consequences

The properties in Definition 2 are sufficient for establishing NP-hardness of CSP

(B

∨=

)

, and it is thus natural to ask to whichextent they are alsonecessary. Althougha complete answerseems difficult toobtain, we mayatleastobserve that if

≺ ∈

B

isan acyclicordersuch that every stricttotal orderinit contains atmostk elements andthereis atleast one strict total order with three or more elements, then CSP

(

B

∨=

)

is NP-hard, regardless of whether

≺

has properties C2–C4 ornot.This canbe seenvia apolynomial-timereduction fromk-Colourability (i.e.theproblemCSP

(

{

Rk

})

where Rk

= {(

x,y)

∈ {

1

,

. . . ,

k

}

2

|

x

=

y

}

)toCSP

(B

∨=

)

.Let

(V

,

E)beanarbitraryundirectedgraph.Introducevariablesc1

,

. . . ,

ckfor eachcolour,andconstrainthemasc1

(

≺)

c2

(

≺)

. . . (

≺)

ck.Foreachvertexv

∈

V ,introduceavariable w andtheconstraints w(

≺,

,

eqD

)c

i, 1

≤

i

≤

k. Recall that

,

eqD

∈

B

since

B

is apartition schemeso the relation

(

≺,

,

eqD

)

is a member of

B

∨=.Notethattheseconstraintsimplythat w equalsexactly onecolourvariablein anysatisfyingassignment. Finally, introducetheconstraintw(

≺,

)

wforeachedge

(v,

v

)

inE.ItiseasytoverifythattheresultingCSP

(

B

∨=

)

instancehasa solutionifandonlyif

(V

,

E

)

isk-colourable.Itisalsoeasytoverifythatthereductioncanbecomputedinpolynomialtime sincek isaconstantthatonlydependsonthechoiceof

B

.Sincek-Colourability isNP-hardwheneverk

≥

3,NP-hardness ofCSP

(

B

∨=

)

follows.

Similarly,itisnaturaltoaskwhathappensif

≺

isan acyclicorderthatcontainsinfinitestricttotalordersbutdoesnot havein- and/orout-forks.Wehaveseenthatthissometimesleadstotractability,asinthecaseofe.g.thepointalgebraand thebranchingtimealgebra,butthisisnotalwaysthecase.Forasimplecounterexample,letD

= {(

0

,

i),

(

1

,

i),

(

2

,

i)

|

i

∈ N}

anddefine

≺ ⊆

D2 such that

(a,

b)

≺ (

c,d) ifandonly ifa

=

c and b

<

d.It is easy toverify that

≺

isan acyclic order (in fact,itis astrict partialorder),it contains infinitestrictpartial orders(suchas

(

0

,

0

)

≺ (

0

,

1

)

≺ (

0

,

2

)

≺ . . .

),andthat it doesnothave in- orout-forks.Let

B = {≺,

,

,

eqD

}

where



=

D2

\



{≺,

,

eqD

}

,andobservethat

B

is apartition scheme.WeshowthatCSP

(

B

∨=

)

isanNP-hardproblemviaapolynomial-timereductionfrom3-Colourability. Let

(V

,

E

)

be anarbitraryundirectedgraph.Foreachvertexv

∈

V ,introduceavariable w,andforeachedge

(w,

w

)

∈

E, introduce theconstraintw



w.Notethat

((a,

b),

(c,

d))

∈ 

ifandonlyifa

=

c andthata andc arerestrictedtothethree-element set

{

0

,

1

,

2

}

. Giventhis, it is easy to verifythat the resulting CSP

(

B

∨=

)

instance has a solution ifand only if

(V

,

E

)

is 3-colourable.

4.3. Atractablesubclassofdegree-boundedproblems

GiventhatCSP

(B

∨=

)

-3 isNP-hardwhenever

B ∈ H

,itisinterestingtoseewhetherthedegreeboundcanbefurther low-eredwithretainedNP-hardnessornot.Wewillnotbeabletoanswerthisquestioninitsfullgeneralitybutfor

ω

-categorical

partition schemes

B

,CSP

(B

∨=

)

-2 is, to thecontrary, always solvableinpolynomial time. Webeginby recapitulatingthe conceptof

ω

-categoricityanditsconnectionstoqualitativeCSPs.Afirst-ordertheory isasetoffirst-ordersentencesandthe first-ordertheoryofaconstraintlanguage



isthesetoffirst-ordersentencesthatarelogicallyentailedby



.Wesaythat asatisfiable first-ordertheory T is

ω

-categoricalifallcountablemodelsof



areisomorphic,anda constraintlanguage is

ω

-categorical ifitsfirst-ordertheoryis

ω

-categorical.ItisknownthatalargenumberofqualitativeCSPscanbecapturedvia

ω

-categorical constraintlanguages:well-knownexamplesincludeAllen’salgebra [21] andRCC-8 [8].Manymoreexamples canbefoundinSection1ofBodirsky&Jonsson [7].

Thebasicideabehindourtractabilityresultistostudythetree-widthofCSP

(

B

∨=

)

-2 instancesandexploit aresultby Bodirsky & Dalmau [6].A tree-decomposition of a graph G

= (

V

,

E) is a pair

D = (T ,

f

)

, where

T

is a rooted treewith vertexsetT and f

:

T

→

2V isafunctionsuchthatthefollowingpropertieshold:

(T1)



t∈T f

(t)

=

V ,

(T2) foreveryu

∈

V ,theset

{

t

∈

T

|

u

∈

f

(t)

}

inducesaconnectedsubtreeof

T

,and

(T3) foreachedge

(u,

v)

∈

E,thereexistsat

∈

T suchthat

{

u,v

}

⊆

f

(t)

.

The width ofthe tree-decomposition

D

is maxt∈T

|

f

(t)

|

−

1, andthe treewidth of G is the minimum width over all tree-decompositionsofG.

We willconsiderthetreewidthofGaifmangraphs.TheGaifmangraph(ortheprimalgraph)ofaCSPinstance

(V

,

C) is thegraphonvertexsetV wheretwodistinctverticesvi andvjareadjacentifandonlyifviandvjsimultaneouslyappear inthescopeofsomeconstraintinC .ThefollowingresultisadirectconsequenceofCorollary 1inBodirsky&Dalmau [6].

Proposition11.Let



beafinite

ω

-categoricalconstraintlanguage.ThenCSP()restrictedtoinstanceswhoseGaifmangraphshave tree-widthboundedbysomeconstantissolvableinpolynomialtime.

Theorem12.If

B

isan

ω

-categoricalpartitionscheme,thenCSP(

B

∨=

)-2 is

solvableinpolynomialtime.

Proof. If

B

is

ω

-categorical,then

B

∨= is

ω

-categorical,too,since thisproperty ispreservedunderfirst-order definitions [22,Theorem7.3.8]. Let

(V

,

C)bean arbitraryinstanceofCSP

(B

∨=

)

-2.Sincetherelationsin

B

∨= arebinary,itiseasy to seethat the Gaifmangraph of

(V

,

C)isthe disjointunionofsimple pathsandcycles.It iswell-known (andnotdifficult to verify)thatsuch a graphhastree-widthatmost2.TheresultfollowsfromProposition 11since

B

∨= isafinite setof relations.



5. Fasterexponential-timealgorithmsforCSP

(

B

∨=

)

WehaveestablishedthatCSP

(B

∨=

)

for

B ∈ H

isunlikelytobesolvableinsubexponentialtime,sowefocusourefforts on constructingfasterexponential-time algorithms.Algorithm 1(thatwe presentbelow) isan abstractdescriptionofthe “classical” backtracking algorithm forsolving CSPsoverpartition schemes. It isnot so difficultto seethat thisalgorithm solvesCSP

(B

∨=

)

in2O(n2)time(cf.Jonsson&Lagerkvist [25] orRenz&Nebel [38]),andwewillsoonseethatthisalgorithm appliedtothedegree-boundedproblemrunsin2O(n)_{time.Thealgorithmiswell-knownbutwegiveadetailedaccountof}

itsinceitmakesthepresentationofthedegree-boundedcasemuchsimpler.Wenotethatthegeneralupperboundcanbe improvedforseveralspecificpartitionschemes,forinstance:

•

theCSPoverAllen’sintervalalgebraadmitsa2O(n log n)_{timealgorithm [25],and}

•

theCSPoverAllen’sintervalalgebrarestrictedtointervalsofunitlengthadmitsa2O(n log log n) _{timealgorithm [11].}

Importantly,aswewill proveinthissection, theCSPproblemoverRCC-8 alsoadmitsa 2O(n log n) time algorithm.This immediatelyimplies,forinstance,thattheCSPsforRCC-5andthepartial-ordertimealgebracanbesolvedwithinthistime bound,too [3].

5.1. Thebranchingalgorithm

Whenpresentingthebranchingalgorithmweforsimplicityassumethattheconstraintlanguage

B

isapartitionscheme where CSP(

B

) is solvable in polynomial time. One can show that an instance I

= (

V

,

C) of CSP

(B

∨=

)

is satisfiable by providingacertificate definedasasatisfiableinstanceI

= (

V

,

C

)

ofCSP(

B

)obtainedbyremovingallbutonerelationfrom eachconstraintinC .Notethat I issatisfiableifandonlyifithasacertificate:asatisfyingassignmenttoI satisfiesI,and viceversa.

Lemma13.Let

B

beapartitionschemewhereCSP(B)issolvableinpolynomialtime.ThenAlgorithm1solvesaninstanceI

= (

V

,

C

)

ofCSP(

B

∨=

)

in2|C|log(|B|−1)

_·

_poly(

_||

_I

_||)

_time.