combinatorial generation problemsperezguillau.me/document/combi_tutorial_2018.pdf · •...

CombinatorialGenerationProblems

GuillaumePerez

Motivations

• Combinatorialproblemsareeverywhere

2

Pickupanddeliveries

DamsplacementinamazonKnapsackproblems

Musicandtextgeneration

CombinatorialOptimizationProblems

• Findinganoptimal objectfromafinitesetofobjects• Example:Findingtheshortestpathinagraph

• Decisionvariablesarediscrete• Differentfromcontinuousoptimization

3

Easywithcontinuous variables Hard withdiscretevariables

max

Pi2S xivi

subject to:

Pi2S xiwi W

Knapsackproblems

xixi

Theend?

ThankYou

• Questions?

4

OnSolvingCombinatorialProblems

5

ProblemModeling

Optimization

SolutionModel


6

ProblemModeling

Optimization

SolutionModel


7

ProblemModeling

Overapproximation

Optimization

SolutionModel


8

ProblemModeling

Overapproximation

Underapproximation

Optimization

Solution

ModelingisanArt,butoptimizingisascience

Model

SequenceGeneration

• Generateasequence

• Butwithconstraints• #orange<3• #apple>1• Finishbystrawberry

9

Productlinescheduling

x1 x2 x3 x4 x5 x6 x7

7 5 3 5 3 7 7

Generationunderconstraints

• WhatifIwanttogenerateatextwithrhymes?

• Textandmusicusingsomeone’s“style”?

• Musicwithoutplagiarism?

• Poetrywithagivennumberofsyllables?

10

FindarhymewithcombinatorialThatdoesn’tsound artificial

Letitgo,letitgoCan'thold itbackanymore

Inthemonthofthe longdeclineofrosesI,beholdingthesummerdeadbeforeme,Setmyfacetotheseaandjourneyedsilent,

Algernon CharlesSwinburne

Examplewithimage






11





Examplewithimage

SequencewithRhymes

• Example,rhymesbetweeneachtwolines

• Howtoexpress that?• 1variablepersyllable?• 1variableperword?

12

IbePuertoRicandayparadefloatin’ThatBenzMarinaDelReycoastin’SheinschooltobearealestateagentLastmonth Ihelpedherwiththecarpayment

Famous– KanyeWest

x1 x2 x3 x4 x5x6 x7 x8 x9 x10

Constraint:x5 andx10 mustrhyme.

https://youtu.be/1wYXSxCvN68?t=73

SequencewithRhymes

• Givenacorpus

• Generatesentencesthatrhyme

• WhatvaluesfortheXi withi ≠{5,10}?

• WhatvaluesfortheXi withi ={5,10}?

13

x1 x2 x3 x4 x5x6 x7 x8 x9 x10


Section2inexercises






14





Examplewithimage

SequencewithStyle

• What’ssomeonestyle?• It’sownwayofdoing orpresentingthings.

• Howtoextract it?

• Howtouseittogenerate newcontents?

15

Blindweare,ifcreationofthisclonearmywecouldnotsee.

Yoda

extract Style

generate

SequencewithStyle

• Example• Extract thewordstransitionprocess

• Generate• Great lords build

16

• Toycorpus:Great lords listenbuteverybourgeois listenwhilegreat bourgeoisbuildwheneverylordsbuild

redcatsfollowbutyellowdogssleepwhilereddogsfollowwhenyellowcatssleep

Section3inexercises

ThisisastateJ

SequencewithStyleandRhymes

• Complexity?• NP-complete (reductionfrom3-SAT)toprovethatitexistsasolution• #P-complete(reductionfromcounting3-SAT)forexactsampling

17

Theend?

ThankYou

• Questions?

18

ModelingWithConstraints

• Manyframeworksexist• Satsolvers• MIPsolvers• ConstraintProgramming(CP)solvers

• Modeling• Variableswithdomain

• Constraints

19

x1 x2 x3 x4 x5x6 x7 x8 x9 x10


BlackboxSolvers


• Ourfirstmodel• 10Variables

• 5bysentence• Domain:allpossiblewordsfromcorpus

• Transitionconstraint• Betweenconsecutivevariables

• Rhymeconstraint• Betweenthetwolastwords

20

X1

X2

X3

X4

X5

X6

X7

X8

X9

X10

Variables

Transitio

n

Transitio

n

Transitio

n

Transitio

n

Transitio

n

Transitio

n

Transitio

nTransitio

n

Transitio

n

Rhym

e

Transitionconstraint?

• Standalone• Justpickonewordinthepossiblenext

• Constraint• Booleanformula:• Table:

21

• Toycorpus:Great lords listenbuteverybourgeois listenwhilegreat bourgeoisbuildwheneverylordsbuild


Xi

Xi+1Transitio

n =great

=lords∨ =bourgeois

Xi Xi+1

great lordsgreat bourgeois

Xi Xi+1

great lordsgreat bourgeois… …

Xi=great∧ Xi+1=lords∨Xi=great∧ Xi+1=bourgeois

Section5inexercises






22





Examplewithimage

SequenceWithoutplagiarism

• Problem:• MaxOrder (Papadopoulosetal.,AAAI2014)• Avoiding plagiarisminMarkovsequencegeneration

• Exampleofgoal:• Basedonacorpus

• Generate sequences

• All thesubsequences ofsize2belongtothecorpus• “Greatlordssleep”Fail(unknowntransition)

• None ofthesubsequences ofsize3belongstothecorpus• “Greatlords listen“Fail(plagiarism)

• “Redcatssleep”Success

23

• Toycorpus:Greatlordslistenbuteverybourgeois listenwhilegreatbourgeoisbuildwheneverylordsbuild



• Ourfirstmodel• 10Variables

• 5bysentence• Domain:allpossiblewordsfromcorpus

• Transitionconstraint• Betweenconsecutive2variables

• Rhymeconstraint• Betweenthetwolastwords

• PlagiarismConstraint• Betweenconsecutive3variables

24

X1

X2

X3

X4

X5

X6

X7

X8

X9

X10

Variables

Transitio

n

Transitio

n

Transitio

n

Transitio

n

Transitio

n

Transitio

n

Transitio

nTransitio

n

Transitio

n

Rhym

e

Forbidden

Forbidden

Forbidden

Forbidden

Forbidden

Forbidden

Forbidden

Forbidden

Section6inexercises

Inconsistency

• Example

• Solution:• Usemodelingtools

25

X1

X2

X3

X4

Constraint

1

Constraint

2

X1 X2 X3 X4

a b a bb a b a

X1 X2 X3 X4

a a a ab b b b

Constraint 1

Constraint 2Nosolution!

Combinatorialproblemsolving

• Needforefficientmodelingtools• Hardtoexpress

• Let’susethesolutions

• Efficientdatastructure:1. Efficientlyrepresentsdiscreteproblemsolutions

1. Compression

2. Efficientlycombinable2. Combination

3. Integratedintosolversviafastandincrementalalgorithms.3. Integration

26

root

a

0

b

1

c

0 2 1

d

02

tt

1 2 0

• BDDgeneralization

• EachLayerrepresentsavariable• Eacharc islabeledbyavalue

• Eachpath fromroot nodetott noderepresentsavalidassignment

• Exponentialcompressionpower• Exponentialnumberortuples

Multi-valuedDecisionDiagram(MDD)

X1

X2

X3

27

root

a

0

b

1

c

0 2 1

d

02

tt

1 2 0

Multi-valuedDecisionDiagram(MDD)

28

X1 X2 X30 0 10 0 20 2 10 2 21 1 11 1 21 0 01 2 0

24valuesVS 10arcs+6nodes

1090 tuplesVS14,000nodes100variables600,000arcs

root

a

0

b

1

c

0 2 1

d

02

tt

1 2 0

Reduction• Operationwhichmerges equivalent nodes

• Twonodesareequivalentiftheyhavethesameoutgoingarcs

• Foragivenvariableordering• Canonical

root

a

0

b

1

c

0 1

d

2

e

0 1

tt

0 1 1 0 1

29

Reduction• Operationwhichmerges equivalent nodes

• Twonodesareequivalentiftheyhavethesameoutgoingarcs

• Foragivenvariableordering• Canonical

• Compressionismadebyreduction• Sub-graphsharing

root

a

0

b

1

ce

0 1

d

2 01

tt

011

30

Combination

root1

a

0

b

1

ce

0 1

d

2 01

tt1

011

root2

k

0 2

l

0 2

tt2

1 2

31

∩ =

root

0

0 2

tt

1

Allsetoperations

GoalMDDTransition∩¬Plagiarism

32



Redcatssleep

r

great every red yellow

lords bourgeois bourgeois lords

tt

build listen

cats dogs cats dogs

sleep follow

MaxOrderMDDmodel

Corpus

M

MarkovMDDm (Size3)

33

MarkovMDD

34



Redcatssleepr


lords bourgeois

tt

build listen

cats dogs

sleep follow

MaxOrderMDDmodel

Corpus

M

P

MarkovMDDm (Size3)

ProhibitedsequencesMDD3 (Size3)

35

ProhibitedMDD

36

• Toycorpus:Greatlordslisten buteverybourgeois listenwhilegreatbourgeoisbuildwheneverylordsbuild

redcatsfollowbutyellowdogssleepwhilereddogsfollowwhenyellowcatsleep

r



tt

listen build

cats dogs cats dogs

follow sleep

MaxOrderMDDmodel

Corpus V

Allthesolutions (size3)

MDDv (valid)

M

P

MarkovMDDm (Size3)

ProhibitedsequencesMDD3 (Size3)

DeletionM– P

Remark:P⊆M

37

Removing

38

r



tt

build listen

cats dogs cats dogs

sleep follow

r


lords bourgeois

tt

build listen

cats dogs

sleep follow

r



tt

listen build

cats dogs cats dogs

follow sleep

=-MarkovMDD ProhibitedMDD

Allsolutions

MaxOrderMDDmodel

V

X1

X2

X3

X4

X5

X6

X7

X8

X9

X10

X11

IntersectionMDDGlobalValid

VariablesConstraints

39

V

V

V

V

V

V

V

V

MaxOrder Results

• MDDcontainsallthesolutions• NolongerneedasolverJ• Canbeembeddedwithotherconstraints

• Benchmark“Proust– Àlarecherchedutempsperdu”• 11,000differentwords(domain)• MaxPlagiarism:4• Generatesequencesoflength20• 2creations&1deletion&16intersections

• MDDSize• 1Mnodes, 190Marcs• 2.2*1035 tuples

• Results• First:~400seconds, equivalent totheadhocalgorithm• Orderingeffect:143s

40

Compressionratio:1027

MDDGlobalValid






41





Examplewithimage

Sequencewithagivennumberofsyllables

• Syllabicpoeticmeter• 12syllables

• Frenchpoetsfrom17th to19th century

• Automaticgenerationfromacorpus• Example• Grands seigneurs, toutbourgeoisveut batir comme unBœuf

• Greatlords,everybourgeoiswantstobuildlikeaBeef• Grands – sei – gneurs - tout– bour – geois – veut – bâ- tir – comme – un– Bœuf• 1 2 3 4 5 6 78910 11 12

42

Sampling Model

Corpus

M

MarkovMDDm (Sizen)

43

MarkovMDD

44


redcatsfollowbutyellowdogssleepwhilereddogsfollowwhenyellowcatsleep

Redcatssleepr


lords bourgeois

tt

build listen

cats dogs

sleep follow

Sampling Model

Corpus

M

MarkovMDDm (Sizen)

M ∑

Arclabelreplacedbytheirnumberofsyllables

45

ValueTransformation

46

Transformationr


lords bourgeois

tt

build listen

cats dogs

sleep follow

r

1 3 1 2

1 2

tt

1 2

1 1

1 2

Sampling Model

Corpus

M

MarkovMDDm (Sizen)

∑

∑(1,k)=12Sumofnvariableshavingthevalues1tokasdomain

M ∑


47

SumMDD

• ∑(1,3)=5• Allthepathsarecomposedofvalueswhosesumis5.

r

1

1

2

2

3

3

2

1

3

2

4

3 1 2 1

tt

3 2 1

48

Sampling Model

Corpus

M

MarkovMDDm (Sizen)

∑

∑(1,k)=12Sumofnvariableshavingthevalues1tokasdomain

M ∑


AIntersection

Allthesolutions

49

Intersection

50

r

1

1

2

2

3

3

2

1

3

2

4

3 1 2 1

tt

3 2 1

r

1 3 1 2

1 2

tt

1 2

1 1

1 2

∩ =r

1

1

3

3

2

2

3

2

4

1

tt

2 1

3

11

2

Conversion

r

1

1

3

3

2

2

3

2

4

1

tt

2 1

3

11

2

• Solutions:• Sequencesofsize2fromthecorpus• Sumofsyllablesequal5• (Usually12)

51

r

1

great

3

every

2

yellow

3

bourgeois

4

lords

tt

listen build

3

cats dogs

follow

Transformation

• NowSample!• BigMDDs

• Millionsofnodes• Hugeamountofsolution• 1090 in our problems

Conclusion

• Learntomodelandsolvesomeproblems

• Usingstandalonemethods

• Usingbasicsofmodelingandsolvers

• Usingsomeadvancemodelingtools

• Providedgoodmodelforsomehardproblems

52

ThankYou

• Questions?• ForrealJ

53

CombinatorialGenerationProblems

GuillaumePerez

combinatorial generation problemsperezguillau.me/document/combi_tutorial_2018.pdf · •...

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