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

Modeling Solution Dominance over CSPs Tias Guns, Peter Stuckey, Guido Tack ModRef 2018

C o n s t r a i n e d s a t i s f a c t i o n a n d o p t i m i s a t i o n C o n s t r a i n t m o 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 o n O p t i m i s a t i o n F i n d a s a t i s f y i n g s o l u t i o n M i n i m i z e / m a x i m i z e o n e o b j e c t i v e ( o r fi n d a l l s a t i s f y i n g s o l u t i o n s ) F i n d a b e s t s o l u t i o n

B e y o n d o p t i m i s a t i o n L e x i c o g r a p h i c o p t i m i s a t i o n  (pareto-frontier solutions) M u l t i - o b j e c t i v e o p t i m i s a t i o n  (solutions with smallest subset of true Boolean variables in set X) X - m i n i m a l m o d e l s  (like MaxSAT) We i g h t e d ( p a r t i a l ) M a x C S P  (each constraint has a value for being satisfjed) V a l u e d C S P  (MSS, MCS, MUS) 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  (expresses preferences through a DAG of conditional preference tables) C P - n e t s  (e.g. in itemset mining: closedness and maximality) D o m a i n s p e c i fi c d o m i n a n c e r e l a t i o n s  → not available in constraint modeling languages!

S o l u t i o n d o m i n a n c e dominance relation A s o l u t i o n s p e c i fi e s w h e n o n e s o l u t i o n d o m i n a t e s a n o t h e r H o w t o f o r m a l i z e t h a t o n e s o l u t i o n d o m i n a t e s a n o t h e r ?

111 P r e - o r d e r 110, refmexive transitive A p r e - o r d e r i s a n d 101, 011 → t h i n k p a r t i a l o 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 100 010 001 E x a m p l e s d o m i n a n c e r e l a t i o n s : 000 O p t i m i s a t i o n ( m i n ) :  M u l t i - o b j e c t i v e o p t i m i s a t i o n :  X(v) is truth value {0,1} of v in X X - m i n i m a l m o d e l s : 

F r o m d o m i n a n c e r e l a t i o n t o s o l u t i o n s e t solution set Wh a t i s t h e o f a C o n s t r a i n e d D o m i n a n c e P r o b l e m ( C D P ) ? (every CSP solution is dominanted or C o m p l e t e C o m p l e t e   equivalent to one of the CDP solution) D o m i n a t i o n - f r e e  (CDP solutions are not D o m i n a t i o n - f r e e  (no two CDP solutions E q u i v a l e n c e - f r e e  dominated by other CDP solutions, except are equivalent to each other) equivalent ones) → t h i s s e t i s N O T u n i q u e → t h i s s e t i s u n i q u e → e q u i v a l e n t s o l u t i o n s a r e t y p i c a l l y n o t o f effjcient → i n M u l t i - O b j e c t i v e o p t i m i s a t i o n , t h i s i s t h e i n t e r e s t set ( e v e n s o i n s t a n d a r d o p t i m i s a t i o n )

D e t a i l e d e x a m p l e : m u l t i - o b j e c t i v e M u l t i - o b j e c t i v e

M o r e e x a m p l e s . . . X - m i n i m a l m o d e l s : → C P - n e t : ● d o m i n a n c e i n t e r m s o 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 o n e ) : N P - h a r d ● c a n p l a y w i t h o t h e r d o m i n a n c e r e l a t i o n s , e . g . l o c a l d o m i n a n c e ( f o r e q u a l p a r e n t s o n l y )

D o 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 o l u t i o 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 o e s n o t e x i s t a s u b s e t t h a t i s a l s o f r e q u e n t → X - m a x i m a l s o l u t i o n s ! C l o s e d f r e q . i t e m s e t s : t h e r e d o e s n o 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 o n d i t i o n a l X - m a x i m a l s o l u t i o n s ! → compatible with arbitrary constraints ( a p o s i t i v e t h i n g i n c o 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]

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 o l v i n g m e t h o d s e . g . m u l t i - o b j e c t i v e , M a x C S P , M U S , . . . B u t d o m a i n - s p e c i fi c o n e s d o 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 o n - b a c k t r a c k a b l e n o g o o d s

M o d e l i n g i n a l a n g u a g e dominance nogoods , We p r o p o s e t o m o d e l r a t h e r t h a n d o m i n a n c e r e l a t i o n s : 1 ) c a n b e u s e d t o s p e c i f y b o 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 invariant 2 ) w e f o u n d i t m o r e i n t u i t i v e t o s p e c i f y a n f o r t h e s e a r c h ( e . g . i n c a s e o f m i n i m i s a t i o n , i f S i s a s o l u t i o n t h e n f ( V ) < f ( S ) f o r a n y f u t u r e s o l u t i o n V )

M o 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 o d e l i n g : a p r i m i t i v e f o r s p e c i f y i n g a d o m i n a n c e n o g o o d S e a r c h : p o s t a ( n o n - b a c k t r a c k a b l e ) c o n s t r a i n t e a c h t i m e a s o l u t i o n i s f o u n d * s o 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 o n [A. Rendl, T. Guns, P. Stuckey, G. Tack. MiniSearch: A solver-independent meta-search language for minizinc, CP 2015]

E x a m p l e e x p e r i m e n t s Constraint dominance problems i declarative solver-independent language n a S o l v e r s : g e c o 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 o d e / o r t o o 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 o x r e s t a r t s  free ordered ) S e a r c h s t r a t e g y : o 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 (

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

E x a m p l e : B i - o 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)

C o n c l u s i o n B e y o n d s a t i s f a c t i o n / o p t i m i s a t i o n : Constraint dominance problems declarative solver-independent language i n a ● f r o m d o m i n a n c e r e l a t i o n t o d o m i n a n c e n o g o o d s ● c a n b e a d d e d t o m o 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 o o m f o r d o m a i n - s p e c i fi c d o m i n a n c e r e l a t i o n s ? ( e x a m p l e s ? )

Modeling Solution Dominance over CSPs Tias Guns, Peter Stuckey, Guido Tack ModRef 2018