On Reducing Maximum Independent Set to Minimum Satis fiabili ty Ale x - PowerPoint PPT Presentation

A. Ignatiev, A. Morgado, and J. Marques-Silva On Reducing MIS to MinSAT / On Reducing Maximum Independent Set to Minimum Satis fiabili ty Ale x e y Ig n a t ie v  , A nton i o Mor gad o  , a n d J o a o M a rqu e s - S il v a  ,   I N E S C-ID / I ST , Li s b on , Portu gal  CA S L / C S I, Un i v e rs i ty C o llege D u bli n , I r ela n d t h I nt e rn a t i on al C on fe r e n ce on T he ory a n d A pp lica t i ons o f S a t i s fiabili ty T e st i n g V ie nn a, A ustr ia J u l y  ,    

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T / Pro ble ms st a t e m e nts MIS : c omput e t he largest num be r o f p ai r w i s e non -c onn ec t ed v e rt ice s i n G . MVC : c omput e t he smallest num be r o f v e rt ice s i n G t ha t a r e i n cide nt to all edge s o f G .

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T / Pro ble ms st a t e m e nts MIS : c omput e t he largest num be r o f p ai r w i s e non -c onn ec t ed v e rt ice s i n G . MVC : c omput e t he smallest num be r o f v e rt ice s i n G t ha t a r e i n cide nt to all edge s o f G . ⇓ Gi v e n G , I ⊆ G i s a n M I S so l ut i on ⇔ G \ I i s a n M V C so l ut i on.

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T / Pro ble ms st a t e m e nts MIS : c omput e t he largest num be r o f p ai r w i s e non -c onn ec t ed v e rt ice s i n G . MVC : c omput e t he smallest num be r o f v e rt ice s i n G t ha t a r e i n cide nt to all edge s o f G . ⇓ Gi v e n G , I ⊆ G i s a n M I S so l ut i on ⇔ G \ I i s a n M V C so l ut i on. MaxClq = MIS

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T / Pro ble ms st a t e m e nts MIS : c omput e t he largest num be r o f p ai r w i s e non -c onn ec t ed v e rt ice s i n G . MVC : c omput e t he smallest num be r o f v e rt ice s i n G t ha t a r e i n cide nt to all edge s o f G . ⇓ Gi v e n G , I ⊆ G i s a n M I S so l ut i on ⇔ G \ I i s a n M V C so l ut i on. MaxClq = MIS MinSAT : c omput e t he smallest num be r o f s i mu l t a n e ous l y satisfied cla us e s i n F . MaxFalse : c omput e t he largest num be r o f s i mu l t a n e ous l y falsified cla us e s i n F .

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T / Pro ble ms st a t e m e nts MIS : c omput e t he largest num be r o f p ai r w i s e non -c onn ec t ed v e rt ice s i n G . MVC : c omput e t he smallest num be r o f v e rt ice s i n G t ha t a r e i n cide nt to all edge s o f G . ⇓ Gi v e n G , I ⊆ G i s a n M I S so l ut i on ⇔ G \ I i s a n M V C so l ut i on. MaxClq = MIS MinSAT : c omput e t he smallest num be r o f s i mu l t a n e ous l y satisfied cla us e s i n F . MaxFalse : c omput e t he largest num be r o f s i mu l t a n e ous l y falsified cla us e s i n F . ⇓ Gi v e n F , M ⊆ F i s a M a x Fal s e so l ut i on ⇔ F \ M i s a M i nS A T so l ut i on.

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T / M I S a n d M a x Fal s e MIS ↔ MaxFalse MVC ↔ MinSAT

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / Ba s ic r ed u c t i on v  v  v  v  v    c  = x  ,      = ¬ x  ,  ∨ x  ,  ∨ x  , ∨ x  , c        = ¬ x  ,  ∨ x  , F = c  = ¬ x  , c          = ¬ x  , ∨ ¬ x  , c   

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / Ba s ic r ed u c t i on v  v  v  v  v    c  = x  ,      = ¬ x  ,  ∨ x  ,  ∨ x  , ∨ x  , c        = ¬ x  ,  ∨ x  , F = c  = ¬ x  , c          = ¬ x  , ∨ ¬ x  , c   

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / Ba s ic r ed u c t i on v  v  v  v  v    c  = x  ,      = ¬ x  ,  ∨ x  ,  ∨ x  , ∨ x  , c        = ¬ x  ,  ∨ x  , F = c  = ¬ x  , c          = ¬ x  , ∨ ¬ x  , c   

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / Ba s ic r ed u c t i on v  v  v  v  v    c  = x  ,      = ¬ x  ,  ∨ x  ,  ∨ x  , ∨ x  , c        = ¬ x  ,  ∨ x  , F = c  = ¬ x  , c          = ¬ x  , ∨ ¬ x  , c   

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / Ba s ic r ed u c t i on v  v  v  v  v    c  = x  ,      = ¬ x  ,  ∨ x  ,  ∨ x  , ∨ x  , c        = ¬ x  ,  ∨ x  , F = c  = ¬ x  , c          = ¬ x  , ∨ ¬ x  , c   

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / Ba s ic r ed u c t i on v  v  v  v  v    c  = x  ,      = ¬ x  ,  ∨ x  ,  ∨ x  , ∨ x  , c        = ¬ x  ,  ∨ x  , F = c  = ¬ x  , c          = ¬ x  , ∨ ¬ x  , c   

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / Ba s ic r ed u c t i on v  v  v  v  v    c  = x  ,      = ¬ x  ,  ∨ x  ,  ∨ x  , ∨ x  , c        = ¬ x  ,  ∨ x  , F = c  = ¬ x  , c          = ¬ x  , ∨ ¬ x  , c    Gi v e n a g r a p h G = ( V , E ) , basic reduction c onstru c ts a f ormu la F w i t h e x ac t l y | V | cla us e s a n d | E | v a r iable s.

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / G r eed y r ed u c t i on v  v  v  v  v    c  = ¬ x      = c  x        = ¬ x  ∨ x  F = c  = ¬ x  c          = ¬ x  ∨ ¬ x  c   

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / G r eed y r ed u c t i on v  v  v  v  v    c  = ¬ x      = c  x        = ¬ x  ∨ x  F = c  = ¬ x  c          = ¬ x  ∨ ¬ x  c   

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / G r eed y r ed u c t i on v  v  v  v  v    c  = ¬ x      = c  x        = ¬ x  ∨ x  F = c  = ¬ x  c          = ¬ x  ∨ ¬ x  c   

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T  / G r eed y r ed u c t i on v  v  v  v  v    c  = ¬ x      = c  x        = ¬ x  ∨ x  F = c  = ¬ x  c          = ¬ x  ∨ ¬ x  c    Gi v e n a g r a p h G = ( V , E ) , greedy reduction c onstru c ts a f ormu la F w i t h e x ac t l y | V | cla us e s a n d � | V | v a r iable s.

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T / V a r iable c omp a t ibili t y or igi n al idea — compatible states i n fi n i t e- st a t e m achi n e s s i mp lifica t i on

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T / V a r iable c omp a t ibili t y or igi n al idea — compatible states i n fi n i t e- st a t e m achi n e s s i mp lifica t i on c omp a t ible v a r iable s ca n r e p lace each ot he r

A . Ig n a t ie v , A . Mor gad o , a n d J . M a r qu e s - S il v a On R ed u ci n g M I S to M i nS A T / V a r iable c omp a t ibili t y or igi n al idea — compatible states i n fi n i t e- st a t e m achi n e s s i mp lifica t i on c omp a t ible v a r iable s ca n r e p lace each ot he r v a r iable c omp a t ibili t y ru le s :  no t a uto l o g y gi v e n a cla us e ¬ x  ∨ x  , v a r iable s x  a n d x  a r e not c omp a t ible