复旦大学大数据学院School of Data Science, Fudan University

DATA130008 Introduction to Artificial Intelligence

Constraint Satisfaction Problems魏忠钰

March 27th,2019

Constraint Satisfaction Problems

Constraint Satisfaction Problems

§ Planning:sequencesofactions§ Thepathtothegoalistheimportantthing

§ Pathshavevariouscosts(e.g.depths)

§ Heuristicsgiveproblem-specificguidance

§ Identification:assignmentstovariables§ Thegoalitselfisimportant,notthepath

§ E.g.Nqueens,Sudoku

§ Allpathsatthesamedepth(forsomeformulations)

§ ConstraintSatisfactionProblems(CSPs)arespecializedforidentificationproblems

Constraint Satisfaction Problem

§ Standardsearchproblems:§ Stateisa“blackbox”:arbitrarydatastructure

§ Goaltestcanbeanyfunctionoverstates

§ Constraintsatisfactionproblems(CSPs):§ Stateisrepresentedasafeaturevector

§ Goaltestisasetofconstraints toidentifytheallowablefeaturevector

§ 2

Feature Vector

§ Wehave§ Asetofkvariables (orfeatures)§ Eachvariablehasadomain ofdifferentvalues.§ Astateisspecifiedbyanassignmentofavalueforeachvariable.§ height={short,average,tall},§ weight={light,average,heavy}

§ Apartialstateisspecifiedbyanassignmentofavaluetosomeofthevariables.

§ Acompletestateisapossiblesolutiontotheproblem.

CSP Examples

Example: Map Coloring

§ Variables:

§ Domains:

§ Constraints:adjacentregionsmusthavedifferentcolors

§ Solutionsareassignmentssatisfyingallconstraints,e.g.:

Implicit:

Explicit:

Example: N-Queens

§ Formulation1:§ Variables:§ Domains:§ Constraints

Example: N-Queens

§ Formulation2:§ Variables:

§ Domains:

§ Constraints:

Implicit:

Explicit:

Constraint Graphs

Constraint Graphs

§ BinaryCSP:eachconstraintrelates(atmost)twovariables§ Binaryconstraintgraph:nodesarevariables,arcsshow

constraints

Example: Cryptarithmetic

§ Variables:

§ Domains:

§ Constraints:

Example: Sudoku

§ Variables:§ Each(open)square

§ Domains:§ {1,2,…,9}

§ Constraints:

9-wayalldiff foreachrow

9-wayalldiff foreachcolumn

9-wayalldiff foreachregion

(orcanhaveabunchofpairwiseinequalityconstraints)

Transform N-nary CSPs to Binary constraints one

§ ShowhowasingleternaryconstraintsuchasA+B=Ccanbeturnedinto3binaryconstraintsbyusingauxiliaryvariables.Youmayassumefinite domains.Nextshowhowconstraintswithmorethan3variablescanbetreatedsimilarly.

§ Step1:IntroduceanewvariableZ.§ Domainisthesetofallpossible3-tuplessatisfyingA+B=C.{(a1,b1,c1),(a2,b2,c2),…}

§ Step2:CreatenewconstraintsbetweenZandthreeoriginalvariablesrespectively(thepositioninthetupleislinkedtoanoriginalvariable).§ < 𝑍, 𝐴 >∈ {(((𝑎1, 𝑏1, 𝑐1)𝑎1), …}§ < 𝑍, 𝐵 >∈ {(((𝑎1, 𝑏1, 𝑐1), 𝑏1), …}§ < 𝑍, 𝐶 >∈ {((𝑎1, 𝑏1, 𝑐1))𝑐1), …}

Varieties of CSPs and Constraints

Varieties of CSPs

§ DiscreteVariables§ Finitedomains

§ SizedmeansO(dn) completeassignments§ Infinitedomains(integers,strings,etc.)

§ E.g.,jobscheduling,variablesarestart/endtimesforeachjob

§ Continuousvariables§ A + B = C(A,B,Carerealnumbers)

Varieties of Constraints

§ VarietiesofConstraints§ Unaryconstraintsinvolveasinglevariable:

§ Binaryconstraintsinvolvepairsofvariables,e.g.:

§ Higher-orderconstraintsinvolve3ormorevariables: e.g.,cryptarithmetic columnconstraints

§ Global Constraints, e.g.: AllDiff()

§ Preferences(softconstraints):§ i.e.appleisbetterthanbanana§ Oftenrepresentablebyacostforeachvariableassignment

Real-World CSPs

§ Assignmentproblems:e.g.,whoteacheswhatclass§ Timetablingproblems:e.g.,whichclassisofferedwhenandwhere?§ Hardwareconfiguration§ Transportationscheduling§ Factoryscheduling§ Circuitlayout§ Faultdiagnosis§ …lotsmore!

Solving CSPs

Standard Search Formulation

§ StandardsearchformulationofCSPs

§ Statesdefinedbythevaluesassignedsofar(partialassignments)§ Initialstate:theemptyassignment,{}§ Successorfunction:assignavaluetoanunassignedvariable§ Goaltest:thecurrentassignmentiscompleteandsatisfiesallconstraints

§We’llstartwiththestraightforward,naïveapproach,thenimproveit

Search Methods

§WhatwouldBFSdo?

§WhatwouldDFSdo?

§Whatproblemsdoesnaïvesearchhave?§ Ordering ofvariables.§ Choicesofvalues.§ Theutilizationofconstraints.

Backtracking Search

Backtracking Search

§ BacktrackingsearchisthebasicuninformedalgorithmforsolvingCSPs

§ Idea1:Onevariableatatime in fixed ordering§ Variableassignmentsarecommutative,sofixordering§ I.e.,[WA=redthenNT=green]sameas[NT=greenthenWA=red]

§ Idea2:Checkconstraintsasyougo§ I.e.consideronlyvalueswhichdonotconflictpreviousassignments§ Mighthavetodosomecomputationtochecktheconstraints§ “Incrementalgoaltest”

§ Depth-firstsearchwiththesetwoimprovements§ iscalledbacktrackingsearch

Backtracking Example

Backtracking Search

• Backtracking=DFS+variable-ordering+fail-on-violation• Whatarethechoicepoints?

Improving Backtracking

§Ordering:§ Whichvariableshouldbeassignednext?§ Inwhatordershoulditsvaluesbetried?

§ Filtering:Canwedetectinevitablefailureearly?

§ Structure:Canweexploittheproblemstructure?

Ordering

Ordering: Minimum Remaining Values

§ VariableOrdering:Minimumremainingvalues(MRV):§ Choosethevariablewiththefewestlegalleftvaluesinitsdomain

§Whyminratherthanmax?§ Alsocalled“mostconstrainedvariable”§ “Fail-fast”ordering

Ordering: Least Constraining Value

§ ValueOrdering:LeastConstrainingValue§ Givenachoiceofvariable,choosetheleastconstrainingvalue

§ I.e.,theonethatrulesoutthefewestvaluesintheremainingvariables

§ Notethatitmaytakesomecomputationtodeterminethis!(E.g.,rerunningfiltering)

§Whyleastratherthanmost?

Filtering: Forward Checking

§ Forwardchecking:Crossoffvaluesthatviolateaconstraintwhenaddedtotheexistingassignment

WASANT Q

NSWV

Filtering: Constraint Propagation

§ Forwardcheckingpropagatesinformationfromassignedtounassignedvariables,butdoesn'tprovideearlydetectionforallfailures:

§ NTandSAcannotbothbeblue!§ Constraintpropagation:reasonfromconstrainttoconstraint

WA SA

NT Q

NSW

V

Consistency of A Single Arc

§ AnarcX® Yisconsistent iff foreveryxinthetailthereissomeyintheheadwhichcouldbeassignedwithoutviolatingaconstraint

§ Forwardchecking:Enforcingconsistencyofarcspointingtoeachnewassignment

Deletefromthetail!

WA SA

NT Q

NSW

V

Arc Consistency of an Entire CSP

§ Asimpleformofpropagationmakessureallarcsareconsistent:

§ Important:IfXlosesavalue,neighborsofXneedtoberechecked!§ Arcconsistencydetectsfailureearlierthanforwardchecking§ Canberunasapreprocessororaftereachassignment§ What’sthedownsideofenforcingarcconsistency?

Remember:Deletefromthetail!

WA SANT Q

NSW

V

Enforcing Arc Consistency in a CSP

• Runtime:O(n2d3),canbereducedtoO(n2d2)• How?• Checkexercise6.13formoreinformation.

Limitations of Arc Consistency

§ Afterenforcingarcconsistency:§ Canhaveonesolutionleft§ Canhavemultiplesolutionsleft§ Canhavenosolutionsleft(andnotknowit)

§ Arcconsistencystillrunsinsideabacktrackingsearch!

Whatwentwronghere?

K-Consistency

K-Consistency

§ Increasingdegreesofconsistency

§ 1-Consistency(NodeConsistency):Eachsinglenode’sdomainhasavaluewhichmeetsthatnode’sunaryconstraints

§ 2-Consistency(ArcConsistency):Foreachpairofnodes,anyconsistentassignmenttoonecanbeextendedtotheother

§ K-Consistency:Foreachknodes,anyconsistentassignmenttok-1canbeextendedtothekth node.

§ Higherkmoreexpensivetocompute

Strong K-Consistency

§ Strongk-consistency:alsok-1,k-2,…1consistent

§ Claim:strongn-consistencymeanswecansolvewithoutbacktracking!

§ Why?1. Chooseanyassignmenttoanyvariable2. Chooseanewvariable3. By2-consistency,thereisachoiceconsistentwiththefirst4. Chooseanewvariable5. By3-consistency,thereisachoiceconsistentwiththefirst26. …

§ Lotsofmiddlegroundbetweenarcconsistencyandn-consistency!(e.g.k=3,calledpathconsistency)

§ O(n2d) forasolutiontoStrongK-Consistencyproblem.

Structure

Problem Structure

§ Extremecase:independentsub-problems§ Example:Tasmaniaandmainlanddonotinteract

§ Independentsub-problemsareidentifiableasconnectedcomponentsofconstraintgraph

§ Supposeagraphofnvariablescanbebrokenintosub-problemsofonlycvariables:§ Worst-casesolutioncostisO((n/c)(dc)),linearinn§ E.g.,n=80,d=2,c=20§ 280 =4billionyearsat10millionnodes/sec§ (4)(220)=0.4secondsat10millionnodes/sec

Tree-Structured CSPs

§ Theorem:iftheconstraintgraphhasnoloops,theCSPcanbesolvedinO(nd2)time§ ComparetogeneralCSPs,whereworst-casetimeisO(dn)

Tree-Structured CSPs

§ Algorithmfortree-structuredCSPs:

§ Order:Choosearootvariable,ordervariablessothatparentsprecedechildren

§ Checkthearcconsistencyfromchildtoparent§ Assignvaluefromroottoleaves

Tree-Structured CSPs

§ Algorithmfortree-structuredCSPs:

§ Runtime:O(nd2)(why?)

Tree-Structured CSPs

§ Claim1:Afterbackwardpass,allroot-to-leafarcsareconsistentØEachX®YwasmadeconsistentatonepointandY’sdomaincouldnothavebeenreducedthereafter(becauseY’schildrenwereprocessedbeforeY)

§ Claim2:Ifroot-to-leafarcsareconsistent,forwardassignmentwillnotbacktrack

ØGiven one value of a parent node, there is always a valid value in thechild node to make the assignment consistent.

§ Whydoesn’tthisalgorithmworkwithcyclesintheconstraintgraph?

Improving Structure

Nearly Tree-Structured CSPs

§ Conditioning:instantiateavariable,pruneitsneighbors'domains§ CycleCutset conditioning:instantiate(inallways)asetofvariablessuchthattheremainingconstraintgraphisatree

§ CycleCutset sizecgivesruntimeO((dc)(n-c)d2)

SA

Cutset Conditioning

SA

SA SA SA

Instantiatethecutset(allpossibleways)

ComputeresidualCSPforeachassignment

SolvetheresidualCSPs(treestructured)

Chooseacutset

Cutset Excercise

§ Findthesmallestcyclecutset forthegraphbelow.

Tree Decomposition

§ Idea:createatree-structuredgraphofmega-variables§ Eachmega-variableencodespartoftheoriginalCSP§ Sub-problemsoverlaptoensureconsistentsolutions

M3

{(NT=r,SA=g,Q=b),(NT=b,SA=g,Q=r),…}

Agree:(M1,M2)Î{((WA=g,SA=g,NT=g),(NT=g,SA=g,Q=g)),…}

NSW

SA

¹Q

¹ ¹

Agreeonsharedvars

M1 M2

{(WA=r,SA=g,NT=b),(WA=b,SA=r,NT=g),…}

Agreeonsharedvars

NT

SA

¹WA

¹ ¹

Q

SA

¹NT

¹ ¹

Agreeonsharedvars

M4

V

SA

¹NSW

¹ ¹

Iterative Improvement

Iterative Algorithms for CSPs

§ Localsearchmethodstypicallyworkwith“complete”states,i.e.,allvariablesassigned

§ ToapplytoCSPs:§ Takeanassignmentwithunsatisfiedconstraints§ Operators reassignvariablevalues

§ Algorithm:,§ Startfromacompleteassignment(whilenotasolution)§ Variableselection:randomlyselectanyconflictedvariable§ Valueselection:min-conflictsheuristic:

§ Chooseavaluethatviolatesthefewestconstraints§ I.e.,hillclimbwithh(n)=totalnumberofviolatedconstraints

Example: 4-Queens

• States:4queensin4columns(44 =256states)• Operators:movequeenincolumn• Goaltest:noattacks• Evaluation:c(n)=numberofattacks

Min-Conflicts Algorithm

§ Justlikeahillclimbingapproach

Summary: CSPs

§ CSPsareaspecialkindofsearchproblem:§ Statesarepartialassignments§ Goaltestdefinedbyconstraints

§ Basicsolution:backtrackingsearch

§ Speed-ups:§ Ordering§ Filtering§ Structure

§ Iterativemin-conflictsisofteneffectiveinpractice§～50 formostoftheN-queenproblems

Local Search

§ Systematicsearches:keeppathsin memory,andrememberalternativessosearchcanbacktrack.Solutionisapathtoagoal.

§ Pathmaybeirrelevant,ifonlythefinalconfigurationisneeded(8-queens,ICdesign,networkoptimization,…)

Local Search versus Systematic Search

§ SystematicSearch§ BFS,DFS,IDS,Best-First,A*§ Keepssomehistoryofvisitednodes§ Alwayscompleteforfinitesearchspaces,some

versionscompleteforinfinitespaces§ Goodforbuildingupsolutionsincrementally

§ State=partialsolution§ Action=extendpartialsolution

Local Search versus Systematic Search

§ LocalSearch§ Gradientdescent,Greedylocalsearch,Simulated

Annealing,GeneticAlgorithms§ Doesnotkeephistoryofvisitednodes§ Notcomplete.Maybeabletoarguewill

terminatewith“highprobability”§ Goodfor“fixingup”candidatesolutions

§ State=completecandidatesolutionthatmaynotsatisfyallconstraints

§ Action=makeasmallchangeinthecandidatesolution

Local search

§ HillClimbing§ Simulatedannealing§ Localbeamsearch§ GeneticAlgorithm§ Gradientdecentforcontinuesfunction

Hill-climbing search

§ Generatenearbysuccessorstatestothecurrentstate§ Pickthebestandreplacethecurrentstatewiththatone.

Hill-climbing search

§ Localmaximum:apeakthatislowerthanthehighestpeak,soasuboptimalsolutionisreturned

§ Plateau:theevaluationfunctionisflat,resultinginarandomwalk

§ Ridge:resultinasequenceoflocalmaximathatisverydifficultforgreedyalgorithmstonavigate.

Local maximum Plateau Ridge

States Where Greedy Search Must Succeed

objectivefunctio

n

Target

States Where Greedy Search Might Succeed

objectivefunctio

n

Target

States Where Greedy Search Will NOT Succeed

objectivefunctio

n

Target

Greedy Search Landscape

objectivefunctio

nob

jectivefunctio

n

LocalMinimum

Plateau

Target

Local search

§ HillClimbing§ Simulatedannealing§ Localbeamsearch§ GeneticAlgorithm§ Gradientdecentforcontinuesfunction

Simulated Annealing

Simulated Annealing

§ Combinationofrandomwalkandhillclimbing.§ Generatearandomnewneighborfromcurrentstate.§ Ifit’sbettertakeit.§ Ifit’sworsethentakeitwithsomeprobabilityproportional

tothetemperatureandthedeltabetweenthenewandoldstates.

Simulated Annealing

§ Thealgorithmwandersaroundduringtheearlypartsofthesearch,hopefullytowardagoodgeneralregionofthestatespace

§ Towardtheend,thealgorithmdoesamorefocusedsearch,makingfewbadmoves

Local search

§ HillClimbing§ Simulatedannealing§ Localbeamsearch§ GeneticAlgorithm§ Gradientdecentforcontinuesfunction

Local Beam Search

§ Keeptrackofkstatesratherthanjustone,asinhillclimbing

§ Beginswithkrandomlygeneratedstates§ Ateachstep,allsuccessorsofallkstatesaregenerated§ Ifanyoneisagoal,Done!§ Otherwise,selectsbestksuccessorsfromthecomplete

list,andrepeats

Local Search in Continuous Spaces

initial state vector( ) quantity to optimized

step sizeuntil Goal_Test( ) do

( )

Sf S

SfS S SS

a

a

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=

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negativesteptominimizefpositivesteptomaximizef

Wrap-up

§ Ifyouareinterested,refertochapter4.1– 4.2