Modeling Solution Dominance over CSPs Tias Guns, Peter Stuckey, - - PowerPoint PPT Presentation

SLIDE 1

Modeling Solution Dominance

• ver CSPs

Tias Guns, Peter Stuckey, Guido Tack ModRef 2018

SLIDE 2

C

• n

s t r a i n e d s a t i s f a c t i

• n

a n d

• p

t i m i s a t i

• n

C

• n

s t r a i n t m

• d

e l i n g l a n g u a g e s S a t i s f a c t i

• n

F i n d a s a t i s f y i n g s

• l

u t i

• n

(

• r

fi n d a l l s a t i s f y i n g s

• l

u t i

• n

s ) O p t i m i s a t i

• n

M i n i m i z e / m a x i m i z e

• n

e

• b

j e c t i v e F i n d a b e s t s

• l

u t i

• n

SLIDE 3

B e y

• n

d

• p

t i m i s a t i

• n



L e x i c

• g

r a p h i c

• p

t i m i s a t i

• n



M u l t i

• b

j e c t i v e

• p

t i m i s a t i

• n

(pareto-frontier solutions)



X

• m

i n i m a l m

• d

e l s (solutions with smallest subset of true Boolean variables in set X)



We i g h t e d ( p a r t i a l ) M a x C S P (like MaxSAT)



V a l u e d C S P (each constraint has a value for being satisfjed)



M a x i m a l l y S a t i s fi a b l e s u b s e t s (MSS, MCS, MUS)



C P

• n

e t s (expresses preferences through a DAG of conditional preference tables)



D

• m

a i n s p e c i fi c d

• m

i n a n c e r e l a t i

• n

s (e.g. in itemset mining: closedness and maximality) → not available in constraint modeling languages!

SLIDE 4

S

• l

u t i

• n

d

• m

i n a n c e

A s

• l

u t i

• n

dominance relation s p e c i fi e s w h e n

• n

e s

• l

u t i

• n

d

• m

i n a t e s a n

• t

h e r

H

• w

t

• f
• r

m a l i z e t h a t

• n

e s

• l

u t i

• n

d

• m

i n a t e s a n

• t

h e r ?

SLIDE 5

P r e

• r

d e r

A p r e

• r

d e r i s refmexive a n d transitive → t h i n k p a r t i a l

• r

d e r w i t h e q u i v a l e n c e c l a s s e s E x a m p l e s d

• m

i n a n c e r e l a t i

• n

s :



O p t i m i s a t i

• n

( m i n ) :



M u l t i

• b

j e c t i v e

• p

t i m i s a t i

• n

:



X

• m

i n i m a l m

• d

e l s : X(v) is truth value {0,1} of v in X

000 010 100 001 110, 101, 011 111

SLIDE 6

F r

• m

d

• m

i n a n c e r e l a t i

• n

t

• s
• l

u t i

• n

s e t

Wh a t i s t h e solution set

• f

a C

• n

s t r a i n e d D

• m

i n a n c e P r

• b

l e m ( C D P ) ?



C

• m

p l e t e (every CSP solution is dominanted or equivalent to one of the CDP solution)



D

• m

i n a t i

• n
• f

r e e (CDP solutions are not dominated by other CDP solutions, except equivalent ones) → t h i s s e t i s u n i q u e → i n M u l t i

• O

b j e c t i v e

• p

t i m i s a t i

• n

, t h i s i s t h e effjcient set



C

• m

p l e t e



D

• m

i n a t i

• n
• f

r e e



E q u i v a l e n c e

• f

r e e (no two CDP solutions are equivalent to each other) → t h i s s e t i s N O T u n i q u e → e q u i v a l e n t s

• l

u t i

• n

s a r e t y p i c a l l y n

• t
• f

i n t e r e s t ( e v e n s

• i

n s t a n d a r d

• p

t i m i s a t i

• n

)

SLIDE 7

D e t a i l e d e x a m p l e : m u l t i

• b

j e c t i v e

M u l t i

• b

j e c t i v e

SLIDE 8

M

• r

e e x a m p l e s . . .

X

• m

i n i m a l m

• d

e l s : → C P

• n

e t :

• d
• m

i n a n c e i n t e r m s

• f

p r e f e r e n c e r a n k i n g ( t h e t y p i c a l

• n

e ) : N P

• h

a r d

• c

a n p l a y w i t h

• t

h e r d

• m

i n a n c e r e l a t i

• n

s , e . g . l

• c

a l d

• m

i n a n c e ( f

• r

e q u a l p a r e n t s

• n

l y )

SLIDE 9

D

• m

a i n s p e c i fi c e x a m p l e s . . .

F r e q u e n t i t e m s e t m i n i n g : fi n d a l l s

• l

u t i

• n

s X w h e r e f r e q ( X , D ) > = V a l u e M a x i m a l f r e q . i t e m s e t s : t h e r e d

• e

s n

• t

e x i s t a s u b s e t t h a t i s a l s

• f

r e q u e n t → X

• m

a x i m a l s

• l

u t i

• n

s ! C l

• s

e d f r e q . i t e m s e t s : t h e r e d

• e

s n

• t

e x i s t a s u b s e t t h a t h a s t h e s a m e f r e q u e n c y → c

• n

d i t i

• n

a l X

• m

a x i m a l s

• l

u t i

• n

s ! → compatible with arbitrary constraints ( a p

• s

i t i v e t h i n g i n c

• n

s t r a i n e d i t e m s e t m i n i n g )

Specifically for itemset mining studied in: [B. Negrevergne, A. Dries, T. Guns, S. Nijssen, Dominance programming for itemset mining, ICDM 2013]

SLIDE 10

S e a r c h

S p e c i fi c s e t t i n g s h a v e s p e c i fi c , e ffi c i e n t , s

• l

v i n g m e t h

• d

s e . g . m u l t i

• b

j e c t i v e , M a x C S P , M U S , . . . B u t d

• m

a i n

• s

p e c i fi c

• n

e s d

• n

' t . G e n e r a l s e a r c h m e c h a n i s m ? → i n c r e m e n t a l l y a d d n

• n
• b

a c k t r a c k a b l e n

• g
• d

s

SLIDE 11

M

• d

e l i n g i n a l a n g u a g e

We p r

• p
• s

e t

• m
• d

e l dominance nogoods, r a t h e r t h a n d

• m

i n a n c e r e l a t i

• n

s : 1 ) c a n b e u s e d t

• s

p e c i f y b

• t

h e q u i v a l e n c e

• f

r e e a n d w i t h e q u i v a l e n c e s 2 ) w e f

• u

n d i t m

• r

e i n t u i t i v e t

• s

p e c i f y a n invariant f

• r

t h e s e a r c h ( e . g . i n c a s e

• f

m i n i m i s a t i

• n

, i f S i s a s

• l

u t i

• n

t h e n f ( V ) < f ( S ) f

• r

a n y f u t u r e s

• l

u t i

• n

V )

SLIDE 12

M

• d

e l i n g a n d s e a r c h i n M i n i Z i n c

M

• d

e l i n g : a p r i m i t i v e f

• r

s p e c i f y i n g a d

• m

i n a n c e n

• g
• d

S e a r c h : p

• s

t a ( n

• n
• b

a c k t r a c k a b l e ) c

• n

s t r a i n t e a c h t i m e a s

• l

u t i

• n

i s f

• u

n d * s

• l

v e s e a r c h = M i n i S e a r c h e x t e n s i

• n

[A. Rendl, T. Guns, P. Stuckey, G. Tack. MiniSearch: A solver-independent meta-search language for minizinc, CP 2015]

SLIDE 13

E x a m p l e e x p e r i m e n t s

Constraint dominance problems i n a declarative solver-independent language S

• l

v e r s :



g e c

• d

e

• a

p i w i t h m i n i s e a r c h i n c r e m e n t a l A P I



g e c

• d

e /

• r

t

• l

s / c h u ff e d w i t h m i n i s e a r c h b l a c k b

• x

r e s t a r t s S e a r c h s t r a t e g y : free

• r

s u c h t h a t p r e f e r r e d a s s i g n m e n t s a r e e n u m e r a t e d fi r s t (

• rdered)

SLIDE 14

E x a m p l e : M a x C S P

Providing a guiding search strategy often helps, but not always! Different solvers behave quite differently, can compare thanks to solver-independence

SLIDE 15

E x a m p l e : B i

• b

j e c t i v e T S P

• Shows number of intermediate solutions (not final frontier size)
• Top-rows: free search, bottom-rows: max regret search → search strategy helps
• Oscar has efficient global bi-objective constraint (only relevant in free search)

SLIDE 16

C

• n

c l u s i

• n

B e y

• n

d s a t i s f a c t i

• n

/

• p

t i m i s a t i

• n

: Constraint dominance problems i n a declarative solver-independent language

• f

r

• m

d

• m

i n a n c e r e l a t i

• n

t

• d
• m

i n a n c e n

• g
• d

s

• c

a n b e a d d e d t

• m
• d

e l i n g l a n g u a g e s → c r e a t e s b r e a t h i n g r

• m

f

• r

d

• m

a i n

• s

p e c i fi c d

• m

i n a n c e r e l a t i

• n

s ? ( e x a m p l e s ? )

SLIDE 17

Modeling Solution Dominance

• ver CSPs

Tias Guns, Peter Stuckey, Guido Tack ModRef 2018