16 December 2005 Foundations of Logic and Constraint Programming 1 Constraint (Logic) Programming An overview Heuristics Search Network Characteristics

16 December 2005 Foundations of Logic and Constraint Programming 1

Constraint (Logic) Programming

Anoverview

• HeuristicsSearch

• NetworkCharacteristics

• StaticandDynamicHeuristics

• HeuristicsinSICStus

• AlternativestoPureBacktracking

[Dech04]RinaDechter,ConstraintProcessing,


Heuristic Search

Algorithmsthatmaintainsomeformofconsistency,remove(many?)redundant

valuesbut,notbeingcomplete,donoteliminatetheneedforsearch.

Evenwhenaconstraintnetworkisconsistent,enumerationissubjecttofailure.

Infact,aconsistentconstraintnetworkmaynotevenbesatisfiable(neithera

satisfiableconstraintnetworkisnecessarilyconsistent).

All that is guaranteed by maintaining some type of consistency is that the

networksareequivalent.

Solutions are not “lost” in the reduced network, that despite having less

redundantvalues,hasallthesolutionsoftheformer.


Heuristic Search

Hence,thedomainpruningdoesnoteliminateingeneraltheneedforsearch.

The search space is usually organised as a tree, and the search becomes

someformoftreesearch.

As usual, the various branches down from one node of the search tree

correspond to the assignment of the different values in the domain of a

variable.

Assuch,a tree leafcorresponds toacompletecompound label (includingall

theproblemvariables).

A depth first search in the tree, resorting to backtracking when a node

corresponds to a dead end (unsatisfiability), corresponds to an incremental

completionofpartialsolutionsuntilacompleteoneisfound.


Heuristic Search

Given theexecutionmodelofconstraint logicprogramming (oranyalgorithm

thatinterleavessearchwithconstraintpropagation)

Problem(Vars):-

Declaration of Variables and Domains,

Specification of Constraints,

Labelling of the Variables.

the enumeration of the variables determines the shape of the search tree,

sincethenodesthatarereacheddependontheorder inwhichvariablesare

enumerated.

Takeforexampletwodistinctenumerationsofvariableswhosedomainshave

differentcardinality,e.g.X in 1..2, Y in 1..3andZ in 1..4.


Heuristic Search

enum([X,Y,Z]):-

indomain(X)

propagation

indomain(Y),

propagation,

indomain(Z).

#ofnodes=32

(2+6+24)

1 2 3 4 1 2 3 4 1 2 3 4

1 2 3

1

1 2 3 4 1 2 3 4 1 2 3 4

1 2 3

2


Heuristic Search

enum([X,Y,Z]):- indomain(Z), propagation indomain(Y), propagation, indomain(X).

#ofnodes=40(4+12+24)

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2 3 1 2 312 3 1 2 31

1 2 3 4


Heuristic Search

Theorder inwhichvariablesareenumeratedmayhavean important impact

ontheefficiencyofthetreesearch,since

The number of internal nodes is different, despite the same number of

leaves,orpotentialsolutions, #Di.

Failures can be detected differently, favouring some orderings of the

enumeration.

Depending on the propagation used, different orderings may lead to

differentpruningsofthetree.

Theordering of the domains has no direct influence on the search space,

althoughitmayhavegreatimportanceinfindingthefirstsolution.


Variable Selection Heuristics

To control the efficiency of tree search one should in principle adopt

appropriateheuristicstoselect

Thenextvariabletolabel

Thevaluetoassigntotheselectedvariable

Sinceheuristics forvalue choicewillnotaffect thesizeof thesearchtreeto

be explored, particular attention will be paid to the heuristics for variable

selection.Here,twotypesofheuristicscanbeconsidered:

Static - the ordering of the variables is set up before starting the

enumeration,nottakingintoaccountthepossibleeffectsofpropagation.

Dynamic-theselectionofthevariableisdeterminedafteranalysisofthe

problemthatresultedfrompreviousenumerations(andpropagation).


Width of a Node

Static heuristics are based on some properties of the underlying constraint graphs,

namelytheirwidthandbandwidth.

Firstthewidthofanodeisdefined-

Definition: Width of a node N, given ordering O):

Given some total ordering, O, of the nodes of a graph, the width of a node N,

induced by ordering OisthenumberoflowerordernodesthatareadjacenttoN.

.

As an example, given any

ordering that is increasing from

the root to the leaves of a tree,

allnodesofthetree(exceptthe

root)havewidth1.

1

5

32

76

4

8 9


Width of a Graph

Definition(Width of a graph G, induced by O):

Givensomeordering,O,ofthenodesofagraph,G,thewidthofGinducedby

orderingO,isthemaximumwidthofitsnodes,giventhatordering.

Definition(Width of a graph G):

ThewidthofagraphGisthelowestwidthofthegraphinducedbyanyof its

orderingsO.

1

5

32

76

4

8 9

It is apparent from these

definitions, that a tree is a

specialgraphwhosewidthis1.


Graph Width

Example:

Inthegraphbelow,wemayconsidervariousorderingsofitsnodes,inducing

differentwidths.Thewidthofthegraphis3(thisis,forexamplethewidth

inducedbyorderingO1dir).

2 3

5 6

1

7

4

1 2 3 4 5 6 7

<<wO1dir=3(nodes4,5,6and7)>>wO1inv=5(nd1)

<<wO2dir=5(node1)>>wO1inv=6(nd4)

4 3 2 5 6 7 1


Graph Width and k-Consistency

Asshownbefore,togetabacktrackfreesearchinatree,itwouldbeenough

to guarantee that, for each node N, all of its children (adjacent nodes with

higherorder)havevaluesthatsupportthevaluesinthedomainofnodeN.

Thisresultcanbegeneralisedforarbitrarygraphs,givensomeorderingofits

nodes.

If, according to some ordering, the graph has widthw, and if enumeration

follows that ordering, backtrack free search is guaranteed if the network is

stronglyk-consistent(withk > w).

Infact,likeforthecaseofthetrees,itwouldbeenoughtomaintainsomekind

of directed strong consistency, if the labelling of the nodes is done in

“increasing”order.



Example:

WithorderingLo1dir,strong4-consistencyguaranteesbacktrackfreesearch.

Such consistency guarantees that any 3-compound label that satisfies the

relevant constraints, may be extended to a 4th variable that satisfies the

relevantconstraints.

Infact,anyvaluev1fromthedomainofvariable1maybeselected.Ifstrong

4-consistency ismaintained, values from thedomainsof theother variables

willpossiblyberemoved,butnovariablewillhaveitsdomainemptied.

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5 6

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7

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Example (cont.):

Sincetheorderinginduceswidth3,everyvariableXkconnectedtovariableX1is

connectedatmostwithother2variables“lower” thanXk.Forexample,variable

X6,connectedtovariableX1 isalsoconnectedto lowervariablesX3andX4 (the

sameappliestoX5-X4-X1andX4-X3-X2-X1)

Hence,ifthenetworkisstrongly4-consistent,andifthelabelX1-v1wasthere,this

meansthatany3-compoundlabel{X1-v1,X3-v3,X4-v4}couldbeextendedtoa4-

compound label {X1-v1, X3-v3, X4-v4, X6-v6} satisfying the relevant constraints.

When X1 is enumerated to v1, values v3, v4, v6 (at least) will be kept in their

variabledomains.

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5 6

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7

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Thisexampleshowsthat,aftertheenumerationofthe“lowest”variable(inan

ordering that induces a width w to the graph) of a strongly k-consistent

network(andk>w)theremainingvaluesstillhavevaluesintheirdomains(to

avoidbacktracking).

Being strongly k-consistent, all variables connected to node 1 (variableX1)

areonlyconnectedtoatmostw-1(<k)othervariableswithlowerorder.

Hence, ifsomevaluev1was in thedomainofvariableX1, thenforall these

sets of w variables, some w-compound label {X1-v1,...,Xw-1-vw-1} could be

extendedwithsomelabelXw-vw.

Therefore,whenenumeratingX1=v1theremovaloftheothervaluesofX1still

leaves the w-compound label {X1-v1, ..., Xw-vw} to satisfy the relevant

constraints.



Example (cont.):

The process can proceed recursively, by successive enumeration of the

variables in increasing order. After each enumeration the networkwill have

onevariablelessandvaluesmighteventuallyberemovedfromthedomainof

theremainingvariables,ensuringthatthesimplifiednetworkremainsstrongly

4-consistent.

Eventually,anetworkwithonly4variablesisreached,andanenumerationis

obviouslypossiblewithouttheneed,ever,ofbacktracking.

1 2 3 4 5 6 72 3

5 6

1

7

4



Theideasillustratedinthepreviousexamplecouldleadtoaformalproof(by

induction)ofthefollowing

Theorem:

Any constraint network strongly k-consistent,maybeenumeratedbacktrack

free if there isanorderingOaccording towhich theconstraintgraphhasa

widthlessthenk.

Inpractice,ifsuchanorderingOexists,theenumerationisdoneinincreasing

orderofthevariables,maintainingthesystemk-consistentafterenumeration

ofeachvariable.

Thisresultisparticularlyinterestingforlargeandsparsenetworks,wherethe

added cost of maintaining say, path-consistency (polinomial) may

compensatedbynotincurringinbacktracking(exponential).


MWO Heuristics

Strong k-consistency is of course very costly to maintain, in computational

terms,andthisisusuallynotdone.

Nevertheless,andspeciallyinconstraintnetworkswithlowdensityandwhere

the widths of the nodes vary significantly with the orderings used, the

orderingsleadingtolowergraphwidthsmaybeusedtoheuristicallyselectthe

variabletolabel.Specifically,onemaydefinethe

MWO Heuristics(Minimum Width Ordering):

The Minimum Width Ordering heuristics suggests that the variables of a

constraintproblemareenumerated,increasingly,insomeorderingthatleads

toaminimalwidthoftheprimalconstraintgraph.


MWO Heuristics

The definition refers the primal graph of a constraints problem, which

coincideswith the graph for binary constraints (arcs). For n-ary constraints,

the primal graph includes an arc between any variables connected by an

hyper-arcintheproblemhyper-graph.

For example, thegraphbeingusedcouldbe theprimal graphof aproblem

with2quaternaryconstraints(C1245andC1346)and3ternaryconstraints(C123,

C457andC467).

C123 --> arcsa12,a13ea23

C1245 --> arcsa12,a14,a15,a24,a25anda45

C1346 --> arcsa13,a14,a16,a34,a36anda46

C457 --> arcsa45,a47ea57

C467 --> arcsa46,a47ea67

2 3

5 6

1

7

4


MWO Heuristics

TheapplicationoftheMWOheuristcs,requiresthedeterminationoforderings

O leading to the lowest primal constraint graphwidth. The following greedy

algorithmcanbeusedtodeterminesuchorderings.

function sorted_vars(V,C): Node List; if V = {N} then %onlyonevariable sorted_vars <- N else N <- arg Vi min {degree(Vi,C) | Vi in V} %Nisoneofthenodeswithlessneighbours C’<- C \ arcs(N,C) V’<- V \ {N} sorted_vars <- sorted_vars(V’,C’) & N end if end function


MWO Heuristics

- Theordering[1,2,3,4,5,6,7]obtained,leadstoawidth3forthegraph.

2 3

5 6

1

7

4

2 3

5 6

1

4

2 3

5

1

4

2 3

1

4

2 3

1

2

1

Example:

1. In the graph, node 7 has least degree (=3). Onceremoved node 7, nodes 5 and 6 have least degree(=3).

2. Node6isnowremoved.

3. Nodes5,4(degree3),3(degree2)and2(degree1)aresubsequentlyremoved,asshownbelow.


MWO and MDO Heuristics

2 3

5 6

1

7

4

Both the MWO and the MDO heuristic start the enumeration by those

variables with more variables adjacent in the graph, aiming at the earliest

detectionofdeadends.

(noticethatinthealgorithmtodetectminimalwidthordering,thelastvariables

arethosewithleastdegree).

Example:

MWO and MDO orderings are not necessarily

coincident, and may be used to break ties. For

example,thetwoMDOorderings

O1 = [4,1,5,6,2,3,7]

O2 = [4,1,2,3,5,6,7]

inducedifferentwidths(4and3).


Cycle-Cut Sets

TheMWOheuristicisparticularlyusefulwhensomehighlevelofconsistency

ismaintained,asshown in the theoremrelating itwithstrongk-consistency,

whichisageneralisationtoarbitraryconstraintnetworksoftheresultobtained

withconstrainttrees.

However, since such consistency is hard to maintain, an alternative is to

enumeratetheproblemvariablesinorderto,assoonaspossible,simplifythe

constraintnetworkintoaconstrainttree.

From there, a backtrack free search may proceed, provided directed arc

consistency is maintained, which is relatively inexpensive from a

computationalviewpoint

Thisisthebasicideaofcycle-cutsets.


Cycle-Cut Sets

Obviously,oneisinterestedincycle-cutsetswithlowest cardinality.

Consider a constraint network with n variables with domains of size d. If a

cycle-cutsetofcardinalitykisfound,thesearchcomplexityisreducedto

1. atupleintheseknodeswithtimecomplexityO(dk);and

2. maintenanceof(directed)arcconsistency intheremainingtreewithn-k

nodes,withtimecomplexityO(ad2).

Sincea = n-k-1 foratreewithn-knodes,andasumingthatk is“small”, the

totaltimecomplexityisthus

O(n dk+2).


CCS Heuristics

Onemaythusdefinethe

CCS Heuristics(Cycle-Cut Sets)

The Cycle-Cut Sets heuristics suggests that the first variables of a

constraintproblemtobeenumeratedarethosethatformacycle-cutsetwith

leastcardinality.

Unfortunately, there seems to be no good algorithm to determine optimal

cycle-cutsets(i.e.withleastcardinality).

Two possible approximations correspond to use the sorted_vars algorithm

withtheinverseordering,orsimplystartwiththenodeswithmaximumdegree

(MDO).


CCS Heuristics

Example:

Usingalgorithsorted_varsbutselectingfirstthe

nodeswithhighestdegree,wewouldgetthe

followingvariableselection:

1. Removenode4(degree6)andget

2. Removenode1(degree4)andget

2 3

5 6

1

7

4

2 3

5 6

1

7

4

3. Now, inclusion of any of the other nodes (all with degree 2) turns the

constraint graph into a tree. Selecting node 2, we get the cycle-cut set

{1,2,4},whichisoptimal.


Cycle-Cut Sets

Example:

• Enumeranting first variables 1 and 4, the graph

becomes.

• Therefore, addinganyother node to the set {1,4}

eliminates cycle 2-3-6-7-5-2, and turn the

remainingnodesintoatree.

• Sets{1,4,2},{1,4,3},{1,4,5},{1,4,6}and{1,4,7},are

thuscycle-cutsets.

• For example, after enumeration of variables from

the cycle-cut set {1,4,7}, the remaining constraint

graphisatree.

2 3

5 6

1

7

4

2 3

5 6

1

7

4


MBO Heuristics

Definition(Bandwidth of a graph G, induced by O):

GivenatotalorderingOofthenodesofagraph,thebandwidthofagraphG,

inducedbyO,isthemaximumdistancebetweenadjacentnodes.

2 6

3 5

1

4

7

Example:

WithorderingO1, thebandwithofthegraphis5,the

distancebetweennodes1-6and2-7.Ifthechoiceof

variableX6determinesthechoiceofX1,changesof

X1 will only occur after irrelevantly backtracking

variablesX5,X4,X3andX2!!!

1 2 3 4 5 6 7


MBO Heuristics

Definition(Bandwidth of a graph G):

ThebandwidthofagraphGisthelowestbandwidthofthegraphinducedby

anyofitsorderingsO.

Example:

With orderingO2, the bandwidth is 3, the distance

betweenadjacentnodes2/5,3/6and4/7.

Noorderinginducesalowerbandwidthtothegraph.

Therefore,thegraphbandwidthis3!!!

1 2 3 4 5 6 7

2 3

4 6

1

7

5


MBO Heuristics

Theconceptofbandwidthisthebasisofthefollowing

MBO Heuristics(Minimum Bandwidth Ordering)

TheMinimum Width Orderingheuristicssuggeststhatthevariablesofa

constraintproblemareenumerated,increasingly,insomeorderingthatleads

totheminimalbandwidthoftheprimalconstraintgraph.

Example:

TheMBOheuristicsuggeststheuseofanheuristic

succhasO2,thatinducesabandwidthof3.

1 2 3 4 5 6 7

2 3

4 6

1

7

5


MBO Heuristics

The use of the MBO heuristics with constraint propagation is somewhat

problematicsince:

The constraint graphs in which it ismore useful should be sparse and

possessingnonodewithhighdegree. In the lattercase, thedistanceto

thefarthestapartadjacentnodedominates.

Theprincipleexploitedbytheheuristics,avoidirrelevantbacktracking,is

obtained,hopefullymoreefficiently,byconstraintpropagation.

Noefficientalgorithmsexisttocomputethebandwidthforgeneralgraphs

(see[Tsan93],ch.6).


MBO Heuristics

Thesedifficultiesareillustratedbysomeexamples

Difficulty 1:

The constraint graphs in which it is more useful should be sparse and

possessingnonodewithhighdegree. In the lattercase, thedistance to the

farthestapartadjacentnodedominates

Example:

Node 4, with degree 6, determines that the

bandwith may be no less than 3 (for orderings

with3nodesbeforeand3nodesafternode4).

Many orderings exist with this bandwidth.

However if node 3were connected to node 7,

thebandwidthwouldbe4.

5 6

1 2

3

7

4


MBO Heuristics

Difficulty 2:

Irrelevantbacktrackingishandledbyconstraintpropagation.

Example:

AssumethechoiceofX6isdeterminantforthechoice

ofX1.Constraintpropagationwillpossiblyempty the

domainofX6 ifabadchoice ismade in labelingX1,

before(some)variablesX2,X3,X4andX5arelabeled,

thusavoidingtheirirrelevantbacktracking.

2 6

3 5

1

4

7

TheMBOheuristicsisthusmoreappropriatewithbacktrackingalgorithmswithout

constraintpropagation.

1 2 3 4 5 6 7


MBO Heuristics

Difficulty 3:

Noefficientalgorithmsexisttocomputebandwidthforgeneralgraphs.

Ingeneral,lowerwidthscorrespondtolowerbandwidths,butthebestorderings

inbothcasesareusuallydifferent.

A B

C

D

E

F

G

H

C D A B E F G H

<<Width(minimal)=2;Bandwidth=5

<<Width=3; Bandwidth(minimal)=4

C D E A B F G H


Dynamic Heuristics

In contrast to the static heuristics discussed (MWO, MDO, CCS e MBO)

variableselectionmaybedetermineddynamically.

Instead of being fixed before enumeration starts, the variable is selected

taking into account the propagation of previous variable selections (and

labellings).

Inadditiontoproblemspecificheuristics,thereisageneralprinciplethathas

showngreatpotential,thefirst-failprinciple.

The principle is simple: when a problem includes many interdependent

“tasks”,startsolvingthosethataremostdifficult. It isnotworthwastingtime

withtheeasiestones,sincetheymayturntobeincompatiblewiththeresults

ofthedifficultones.


First-Fail Heuristics

Therearemanywaysofinterpretingandimplementingthisgenericfirst-fail

principle.

Firstly,thetaskstoperformtosolveaconstraintsatisfactionproblemmaybe

consideredtheassignmentofvaluestotheproblemvariables.Howto

measuretheirdifficulty?

Enumeratingbyitselfiseasy(asimpleassignment).Whatturnsthetasks

difficultistoassesswhetherthechoiceisviable,afterconstraintpropagation.

Thisassessmentishardtomakeingeneral,sowemayconsiderfeaturesthat

areeasytomeasure,suchas

Thedomainofthevariables

Thenumber of constraints(degree)theyparticipatein.



Thedomainofthevariables

Intuitively,ifvariablesX1/X2havem /m2valuesintheirdomains,andm2 > m1,

itispreferabletoassignvaluestoX1,becausethereislesschoiceavailable!

Inthelimit,ifvariableX1hasonlyonevalueinitsdomain,(m1 = 1),thereisno

possiblechoiceandthebestthingtodoistoimmediatelyassignthevalueto

thevariable.

Anotherwayofseeingtheissueisthefollowing:

Ontheonehand,the“chance”toassignagoodvaluetoX1ishigherthan

thatforX2.

On the other hand, if that value proves to be a bad one, a larger

proportionofthesearchspaceiseliminated.


First-Fail Heuristics: Example

Example:

In the 8 queens problem, where

queensQ1,Q2eQ3,werealready

enumerated,we have the following

domainsfortheotherqueens

Q4 in {2,7,8}, Q5 in {2,4,8}, Q6 in {4}, Q7 in {2,4,8}, Q8 in {2,4,6,7}.

Hence, thebest variable toenumeratenext shouldbeQ6,notQ4 thatwould

followinthe“natural”order.

Inthisextremecaseofsingletondomains,node-consistencyachievespruning

similartoarc-consistencywithlesscomputationalcosts!

1 1

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Thenumber of constraints(degree)ofthevariables

Thisheuristics isbasically theMaximumDegreeOrdering (MDO)heuristics,

but now the degree of the variables is assessed dynamically, after each

variableenumeration.

Clearly,themoreconstraintsavariableisinvolvedin,themoredifficultitisto

assignagoodvaluetoit,sinceithastosatisfyalargernumberofconstraints.

Of course, and like in the case of the domain size, this decision is purely

heuristic.Theeffectof theconstraintsdependsgreatlyon theirpropagation,

whichdependsinturnontheprobleminhand,whichishardtoantecipate.


Problem Dependent Heuristics

In certain typesof problems, theremightbeheuristics speciallyadapted for

theproblemsbeingsolved.

Forexample,inschedulingproblems,wheretheyshouldnotoverlapbuthave

to take place in a certain period of time, it is usually a good heuristic to

“scatter”themasmuchaspossiblewithintheallowedperiod.

This suggests that one should start by enumerating first the variables

correspondingtotasksthatmaybeperformedinthebeginningandtheendof

theallowedperiod,thusallowing“space”fortheotherstoexecute.

Insuchcase,thedynamicchoiceofthevariablewouldtakeintoaccountthe

valuesinitsdomain,namelytheminimumandmaximumvalues.


Mixed Heuristics

Taking intoaccount the featuresof theheuristicsdiscussedso far,onemay

consider the use of mixed strategies, that incorporate some of these

heuristics.

For example the static cycle-cut sets (CCS) heuristics suggests a set of

variablestoenumeratefirst,soastoturntheconstraintgraphintoatree.

Within this cycle-cut set one may use a first-fail dynamic heuristic (e.g.

smallestdomainsize).

Ontheotherhand,evenastaticheuristiclikeMinimalWidthOrdering(MWO),

may be turned “dynamic”, by reassessing the width of the graphs after a

certain number of enumerations, to take into account the results of

propagation.


Value Choice Heuristics

Someformsofassigninglikelihoodsarethefollowing:

Ad hoc choice:

Againinschedulingproblems,onceselectedthevariablewithlowest/highest

values in its domain, the natural choice for the value will be the

lowest/highest,whichsomehow“optimises”thelikelihoodofsuccess.

Lookahed:

Onemay try to antecipate for each of the possible values the likelihood of

success by evaluating after its propagation the effect on an aggregated

indicatoronthesizeofthedomainsnotyetassigned(asdonewiththekappa

indicator),chosingtheonethatmaximisessuchindicator.


Value Choice Heuristics

Optimisation:

Inoptimisationproblems,wherethereissomefunctiontomaximise/minimise,

onemaygetboundsforthatfunctionwhenthealternativevaluesarechosen

forthevariable,orcheckhowtheychangewiththeselectedvalue.

Ofcourse,theheuristicwillchosethevaluethateitheroptimisesthebounds

inconsideration,orthatimprovesthemthemost.

Notice that in this case, the computation of the boundsmay be performed

either before propagation takes place (less computation, but also less

information)oraftersuchpropagation.

.


Heuristics in SICStus

Beingbasedon theConstraintLogicProgrammingparadigm,aprogram in

SICStushasthestructuredescribedbefore

Problem(Vars):-

Declaration of Variables and Domains,

Specification of Constraints,

Labelling of the Variables.

In the labelling of the variablesX1, X2, ..., Xn, of some list Lx, one should

specifytheintendedheuristics.

Although these heuristicsmay be programmed explicitely, there are some

facilitiesthatSICStusprovides,bothforvariable selectionandvalue choice.



The simplest form to specify enumeration is through a buit-in predicate,

labeling/2,where

the1stargumentisalistofoptions,possiblyempty

The2ndargumentisalistLx = [X1, X2, ..., Xn]ofvariablestoenumerate

By default, labeling([ ], Lx) selects variables X1, X2, ..., Xn, from list Lx,

according to their position, “from left to right”. The value chosen for the

variableistheleastvalueinthedomain.

Thispredicatecanbeusedwithnooptionsforstaticheuristics,providedthat

thevariablesaresortedinthelistLxaccordingtotheintendedordering.



Withanempty listofoptions,predicate labeling([ ],L) is in factequivalent to

predicateenumerating(L)below

enumerating([]).

enumerating([Xi|T]):-

indomain(Xi),

enumerating(T).

wherethebuilt-inpredicate,indomain(Xi),chosesvaluesforvariableXiin

increasingorder.

There are other possibilities for user control of value choice. The current

domainofavariable,maybeobtainedwithbuilt-infd_predicatefd_dom/2.For

example

?- X in 1..5, X #\=3, fd_dom(X,D).

D = (1..2)\/(4..5),

X in(1..2)\/(4..5) ?



Usually, it is not necessary to reach this low level of programming, and a

numberofpredefinedoptionsforpredicatelabeling/2canbeused.

Theoptionsof interest forvaluechoice for theselectedvariableareup and

down, with the obviousmeaning of chosing the values from the domain in

increasinganddecreasingorder,respectively.

Hence,toguaranteethatthevalueofsomevariableischosenindecreasing

order without resorting to lower-level fd_predicates, it is sufficient, to call

predicatelabeling/2withoptiondown

labeling([down],[Xi])



Theoptionsof interest for variable selectionare leftmost, min, max, ff, ffc

and variable(Sel)

leftmost-isthedefaultmode.

o Variablesaresimplyselectedbytheirorderinthelist.

min, max - the variablewith the lowest/highest value in its domain is

selected.

o Useful, for example, in many applications of scheduling, as

discussed.

ff, ffc - implements the first-fail heuristics, selection the variablewith a

domain of least size, breaking ties with the number of constraints, in

whichthevariableisinvolved.



variable(Sel)

This is themostgeneralpossibility.Selmustbedefined in theprogram

asapredicate,whoselast3parametersareVars, Selected, Rest.Given

the list of Vars to enumerate, thepredicate should returnSelected as

thevariabletoselect,Restbeingthelistwiththeremainingvariables.

Otherparametersmaybeusedbefore the last3.Forexample, ifoption

variable(includes(5)) isused, thensomepredicate includes/4mustbe

specified,suchas

includes(V, Vars, Selected, Rest)

which should chose, from the Vars list, a variable, Selected, that

includesVinitsdomain.



Noticethatalltheseoptionsofpredicatelabeling/2maybeprogrammedata

lower level, using the adequate primitives available from SICStus for

inspectionofthedomains.TheseReflexivePredicates,namedfd_predicates

include

fd_min(?X, ?Min)fd_max(?X, ?Max)fd_size(?X, ?Size)fd_degree(?X, ?Degree)

withtheobviousmeaning.Forexample,

?- X in 3..8, Y in 1..5, X #< Y,

fd_size(X,S), fd_max(X,M), fd_degree(Y,D).

D = 1, M = 4, S = 2,

X in 3..4, Y in 4..5 ?



Programqueens_fd_hsolvesthenqueensproblem

queens(N,M,O,S,F)

withvariouslabellingoptions:

Variable Selection

Option ff isused.

Variables that are passed to predicate labeling/2 may or may not be

(M=2/1) sorted from the middle to the ends (for example,

[X4,X5,X3,X6,X2,X7,X1,X8])bypredicate

my_sort(2, Lx, Lo).

Value Choice

Rather than starting enumeration from the lowest value, an offset

(parameterO)isspecifiedtostartenumeration“half-way”.


Alternatives to Pure Backtracking

Constraint Logic Programming uses, by default, depth first search withbacktrackinginthelabellingphase.

Despite being “interleaved” with constraint propagation, and the use of

heuristics,theefficiencyofsearchdependscriticallyofthefirstchoicesdone,

namelythevaluesassignedtothefirstvariablesselected.

Backtracking “chronologically”, these values may only change when the

values of the remaining k variables are fully considered (after some O(2k)

time in the worst case). Hence, alternatives have been proposed to pure

depthfirstsearchwithchronologicalbacktracking,namely

Intelligent backtracking,

Iterative broadening,

Limited discrepancy;and

Incremental time-bound search.


Alternatives to Pure Backtracking

In chronological backtracking, when the enumeration of a variable fails,

backtrackingisperformedonthevariablethatimmediatelyprecededit,even if

this variable is not to blame for the failure.

Various techniques for inteligent backtracking, or dependency directed

search, aimat identifying the causesof the failureandbacktrackdirectly to

thefirstvariablethatparticipatesinthefailure.

Somevariantsofintelligentbacktrackingare:

Backjumping ;

Backchecking ;and

Backmarking .


Intelligent Backtracking

Backjumping

Failingthelabelingofavariable,all

variablesthatcausethefailureof

eachofthevaluesareanalysed,and

the“highest”ofthe“least”variables

isbacktracked

In the example, variable Q6, could not be labeled, and backtracking is

performedonQ4,the“highestoftheleast“variablesinvolvedinthefailureofQ6.

All positions of Q6 are, in fact, incompatible with the value of some variable

lowerthanQ4.

1 3 4 2 5 4 5 3 5 1 2 3


Intelligent Backtracking

Backchecking and Backmarking

These techniquesmaybeusefulwhen the testingofconstraintsondifferent

variables is very costly. The key idea is tomemorise previous conflicts, in

ordertoavoidrepeatingthem.

In backchecking, only the assignments that caused conflicts are

memorised.

Embackmarking theassignments thatdidnot causeconflictsarealso

memorised.

The use of these techniqueswith constraint propagation is usually not very

effective (with a possible exception of SAT solvers, with nogood clause

learning), since propagation antecipates the conflicts, somehow avoiding

irrelevantbacktracking.


Iterative Broadening

In iterative broadening it isassigneda limitb, to thenumberof times thata

nodeisvisited(boththeinitialvisitandthosebybacktracking),i.e.thenumber

ofvaluesthatmaybechosenforavariable.Ifthisvalueisexceeded,thenode

anditssuccessorsarenotexploredanyfurther.

• In theexample,assumingb = 2,

the search space prunned is

shadowed.

Of course, if the search fails for a given b, this value is iteratively increased,

hencetheiterativebroadeningqualification.

1 2 3 4 1 2 3 4 1 2 3 4

1 2 3

1


Limited discrepancy

Limited discrepancy assumes that thevaluechoiceheuristicmayonly faila

(small)numberof times. It directs thesearch for the regionswheresolutions

more likely lie, by limiting to some number d the number of times that the

suggestionmadebytheheuristicisnottakenfurther.

• In the example, assuming

heuristicoptionsattheleftand

d=3thesearchspaceprunned

isshadowed.

Again, if the search fails, d may be incremented and the search space is

increasinglyincremented.

1 2 3 4 1 2 3 4 1 2 3 4

1 2 3

1

d = 3d = 2


Incremental Time Bounded Search

InITBS,thegoalissimilartoiterativebroadeningorlimiteddiscrepancy,but

implementeddifferently.

Oncechosenthevaluesofthefirstkvariables,foreachlabel{X1-v1,..., Xk-vk}

searchisallowedforagiventimeT.

Ifnosolutionisfound,anotherlabelingistested.Ofcourse,ifthesearchfails

for a certain value of T, this may be increased incrementally in the next

iterations,guaranteeingthatthesearchspaceisalsoincreassingiteratively.

In all these algorithms (iterative broadening, limited discrepancy and

incremental duration) parts of the search space may be revisited.

Nevertheless, the worst-case time complexity of the algorithms is not

worsened.


Incremental Time Bounded Search

Forexample, inthecaseof incrementaltime-boundsearchif thesuccessive

andfailediterationsincreasethetimelimitbysomefactor,i.e.Tj+1=Tj,the

iterationswilllast

T1+T2+...+Tj

=T(1++ 2+...+j-1)=

=T(1-j)/(1-)Tj

Ifasolutionisfoundintheiterationj+1,then

thetimespentinthepreviousiterationsisTj.

iterationj+1lastsforTjand,inaverage,thesolutionisfoundinhalfthis

time.

Hence,the“wasted”timeisofthesamemagnitudeofthe“useful”timespent

inthesearch(initerationj+1).

Documents

16 December 2005 Foundations of Logic and Constraint Programming 1 Constraint (Logic) Programming An overview Heuristics Search Network Characteristics