Approximation Strategies for Incomplete MaxSAT Saurabh Joshi 1 - PowerPoint PPT Presentation

Approximation Strategies for Incomplete MaxSAT Saurabh Joshi 1 Prateek Kumar 1 Ruben Martins 2 Sukrut Rao 1 1 IIT Hyderabad 2 Carnegie Mellon University SAT+SMT School 2019, IIT Bombay 8th December 2019 भारतीय ूौ�ो�गक� संःथान हैदराबाद Indian Institute of Technology Hyderabad . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . .. . . . . .

w i r i . r i . . . . MaxSAT w Unsat w w w x x r k . x x r Cardinality Constraint x x r k x x r PB Constraint Minimize k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Weighted MaxSAT) . . . . . ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ) ∧ ( x 1 ∨ ¬ x 2 ) ∧ ( ¬ x 1 ∨ ¬ x 2 )

w i r i . r i . . . . MaxSAT w Unsat w w w x x r k . x x r Cardinality Constraint x x r k x x r PB Constraint Minimize k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Weighted MaxSAT) . . . . . ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ) ∧ ( x 1 ∨ ¬ x 2 ) ∧ ( ¬ x 1 ∨ ¬ x 2 )

w i r i . MaxSAT . . . . . . . . . . w . Unsat w w w r i k Cardinality Constraint k PB Constraint Minimize k . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Weighted MaxSAT) ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ) ∧ ( x 1 ∨ ¬ x 2 ) ∧ ( ¬ x 1 ∨ ¬ x 2 ) ( x 1 ∨ x 2 ∨ r 1 ) ∧ ( ¬ x 1 ∨ x 2 ∨ r 2 ) ∧ ( x 1 ∨ ¬ x 2 ∨ r 3 ) ∧ ( ¬ x 1 ∨ ¬ x 2 ∨ r 4 )

w i r i . . . . . . . . . . . MaxSAT . . w Unsat w w w Cardinality Constraint k PB Constraint Minimize k (MaxSAT) . . . . . . . . . . . . . . . . . . . . . . . (Weighted MaxSAT) . . . . ( x 1 ∨ x 2 ) ∧ ( ¬ x 1 ∨ x 2 ) ∧ ( x 1 ∨ ¬ x 2 ) ∧ ( ¬ x 1 ∨ ¬ x 2 ) ( ∑ r i ) ≤ k ( x 1 ∨ x 2 ∨ r 1 ) ∧ ( ¬ x 1 ∨ x 2 ∨ r 2 ) ∧ ( x 1 ∨ ¬ x 2 ∨ r 3 ) ∧ ( ¬ x 1 ∨ ¬ x 2 ∨ r 4 )

w i r i . . . . . . . . . . . . . . . . MaxSAT Unsat Cardinality Constraint k PB Constraint . . . . . . . . . . . . . Minimize k (Weighted MaxSAT) . . . . . . . . . . . ( x 1 ∨ x 2 ) ∧ w 1 ( ¬ x 1 ∨ x 2 ) ∧ w 2 ( x 1 ∨ ¬ x 2 ) ∧ w 3 ( ¬ x 1 ∨ ¬ x 2 ) w 4 ( ∑ r i ) ≤ k ( x 1 ∨ x 2 ∨ r 1 ) ∧ ( ¬ x 1 ∨ x 2 ∨ r 2 ) ∧ ( x 1 ∨ ¬ x 2 ∨ r 3 ) ∧ ( ¬ x 1 ∨ ¬ x 2 ∨ r 4 )

. . . . . . . . . . . . . . . . MaxSAT Unsat r i k Cardinality Constraint PB Constraint . . . . . . . . . . . . . Minimize k (Weighted MaxSAT) . . . . . . . . . . . ( x 1 ∨ x 2 ) ∧ w 1 ( ¬ x 1 ∨ x 2 ) ∧ w 2 ( x 1 ∨ ¬ x 2 ) ∧ w 3 ( ¬ x 1 ∨ ¬ x 2 ) w 4 ( x 1 ∨ x 2 ∨ r 1 ) ∧ ( ¬ x 1 ∨ x 2 ∨ r 2 ) ∧ ( ∑ w i · r i ) ≤ k ( x 1 ∨ ¬ x 2 ∨ r 3 ) ∧ ( ¬ x 1 ∨ ¬ x 2 ∨ r 4 )

For many applications it may be desirable to fjnd a good solution (even if suboptimal) very quickly. That’s where incomplete solvers . . . . . . . . . . . . . . . . Motivation for MaxSAT come into play! Our contributions Weight relaxation based approximation . . . . . . . . . . . . . . . . . . . . . Subproblem minimization based approximation . . . ▶ Operations Research ▶ Logistics ▶ Resource Allocation ▶ Computational Biology ▶ Fault Localization ▶ ... and many more

That’s where incomplete solvers . . . . . . . . . . . . . . . . Motivation for MaxSAT For many applications it may be desirable to fjnd a good solution come into play! Our contributions Weight relaxation based approximation . . . . . . . . . . . . . . . . . . . . . Subproblem minimization based approximation . . . ▶ Operations Research ▶ Logistics ▶ Resource Allocation ▶ Computational Biology ▶ Fault Localization ▶ ... and many more (even if suboptimal) very quickly.

. . . . . . . . . . . . . . . . Motivation for MaxSAT For many applications it may be desirable to fjnd a good solution come into play! Our contributions Weight relaxation based approximation . . . . . . . . . . . . . . . . . . . . Subproblem minimization based approximation . . . . ▶ Operations Research ▶ Logistics ▶ Resource Allocation ▶ Computational Biology ▶ Fault Localization ▶ ... and many more (even if suboptimal) very quickly. That’s where incomplete solvers

. . . . . . . . . . . . . . . . . Motivation for MaxSAT For many applications it may be desirable to fjnd a good solution come into play! Our contributions . . . . . . . . . . . . . . . . . . . . . . . ▶ Operations Research ▶ Logistics ▶ Resource Allocation ▶ Computational Biology ▶ Fault Localization ▶ ... and many more (even if suboptimal) very quickly. That’s where incomplete solvers ▶ Weight relaxation based approximation ▶ Subproblem minimization based approximation

. l . . . . . . . . . . GTE for Pseudo-Boolean Constraints a . l a l l a o o o o Worst case exponential size (e.g., weights ) . . . . . . . . . . . . . . . . . Polynomial size encoding when all the weights are same. . . . . . . . . . . . ( O : o 2 , o 3 , o 5 , o 6 , o 8 , o 9 , o 11 : 11) ( A : a 2 , a 3 , a 5 : 5) ( B : b 3 , b 6 : 6) ( C : l 1 : 2) ( D : l 2 : 3) ( E : l 3 : 3) ( F : l 4 : 3) ▶ Encoding 2 l 1 + 3 l 2 + 3 l 3 + 3 l 4

. . . . . . . . . . . . GTE for Pseudo-Boolean Constraints . l a l l a o o o o Worst case exponential size (e.g., weights ) . . . . . . . . . . . . . . . Polynomial size encoding when all the weights are same. . . . . . . . . . . . . ( O : o 2 , o 3 , o 5 , o 6 , o 8 , o 9 , o 11 : 11) ( A : a 2 , a 3 , a 5 : 5) ( B : b 3 , b 6 : 6) ( C : l 1 : 2) ( D : l 2 : 3) ( E : l 3 : 3) ( F : l 4 : 3) ▶ Encoding 2 l 1 + 3 l 2 + 3 l 3 + 3 l 4 ▶ ( ¬ l 1 ∨ a 2 )

. . . . . . . . . . . . GTE for Pseudo-Boolean Constraints . l a l l a o o o o Worst case exponential size (e.g., weights ) . . . . . . . . . . . . . . . Polynomial size encoding when all the weights are same. . . . . . . . . . . . . ( O : o 2 , o 3 , o 5 , o 6 , o 8 , o 9 , o 11 : 11) ( A : a 2 , a 3 , a 5 : 5) ( B : b 3 , b 6 : 6) ( C : l 1 : 2) ( D : l 2 : 3) ( E : l 3 : 3) ( F : l 4 : 3) ▶ Encoding 2 l 1 + 3 l 2 + 3 l 3 + 3 l 4 ▶ ( ¬ l 1 ∨ a 2 )

. . . . . . . . . . . . . . GTE for Pseudo-Boolean Constraints l l a o o o o Worst case exponential size (e.g., weights ) . . . . . . . . . . . . . . Polynomial size encoding when all the weights are same. . . . . . . . . . . . . ( O : o 2 , o 3 , o 5 , o 6 , o 8 , o 9 , o 11 : 11) ( A : a 2 , a 3 , a 5 : 5) ( B : b 3 , b 6 : 6) ( C : l 1 : 2) ( D : l 2 : 3) ( E : l 3 : 3) ( F : l 4 : 3) ▶ Encoding 2 l 1 + 3 l 2 + 3 l 3 + 3 l 4 ▶ ( ¬ l 1 ∨ a 2 ) ∧ ( ¬ l 2 ∨ a 3 )

. . . . . . . . . . . . . . GTE for Pseudo-Boolean Constraints l l a o o o o Worst case exponential size (e.g., weights ) . . . . . . . . . . . . . . Polynomial size encoding when all the weights are same. . . . . . . . . . . . . ( O : o 2 , o 3 , o 5 , o 6 , o 8 , o 9 , o 11 : 11) ( A : a 2 , a 3 , a 5 : 5) ( B : b 3 , b 6 : 6) ( C : l 1 : 2) ( D : l 2 : 3) ( E : l 3 : 3) ( F : l 4 : 3) ▶ Encoding 2 l 1 + 3 l 2 + 3 l 3 + 3 l 4 ▶ ( ¬ l 1 ∨ a 2 ) ∧ ( ¬ l 2 ∨ a 3 )

. . . . . . . . . . . . . . . GTE for Pseudo-Boolean Constraints o o o o Worst case exponential size (e.g., weights ) . . . . . . . . . . . . . . . . . . . . . . Polynomial size encoding when all the weights are same. . . . ( O : o 2 , o 3 , o 5 , o 6 , o 8 , o 9 , o 11 : 11) ( A : a 2 , a 3 , a 5 : 5) ( B : b 3 , b 6 : 6) ( C : l 1 : 2) ( D : l 2 : 3) ( E : l 3 : 3) ( F : l 4 : 3) ▶ Encoding 2 l 1 + 3 l 2 + 3 l 3 + 3 l 4 ▶ ( ¬ l 1 ∨ a 2 ) ∧ ( ¬ l 2 ∨ a 3 ) ∧ ( ¬ l 1 ∨ ¬ l 2 ∨ a 5 )

. . . . . . . . . . . . . . . GTE for Pseudo-Boolean Constraints o o o o Worst case exponential size (e.g., weights ) . . . . . . . . . . . . . . . . . . . . . . Polynomial size encoding when all the weights are same. . . . ( O : o 2 , o 3 , o 5 , o 6 , o 8 , o 9 , o 11 : 11) ( A : a 2 , a 3 , a 5 : 5) ( B : b 3 , b 6 : 6) ( C : l 1 : 2) ( D : l 2 : 3) ( E : l 3 : 3) ( F : l 4 : 3) ▶ Encoding 2 l 1 + 3 l 2 + 3 l 3 + 3 l 4 ▶ ( ¬ l 1 ∨ a 2 ) ∧ ( ¬ l 2 ∨ a 3 ) ∧ ( ¬ l 1 ∨ ¬ l 2 ∨ a 5 ) . . .

. . . . . . . . . . . . . . . . . GTE for Pseudo-Boolean Constraints Worst case exponential size (e.g., weights ) . . . . . . . . . . . . . . . . . . Polynomial size encoding when all the weights are same. . . . . . ( O : o 2 , o 3 , o 5 , o 6 , o 8 , o 9 , o 11 : 11) ( A : a 2 , a 3 , a 5 : 5) ( B : b 3 , b 6 : 6) ( C : l 1 : 2) ( D : l 2 : 3) ( E : l 3 : 3) ( F : l 4 : 3) ▶ Encoding 2 l 1 + 3 l 2 + 3 l 3 + 3 l 4 ≤ 5 ▶ ( ¬ l 1 ∨ a 2 ) ∧ ( ¬ l 2 ∨ a 3 ) ∧ ( ¬ l 1 ∨ ¬ l 2 ∨ a 5 ) . . . ¬ o 6 ∧ ¬ o 8 ∧ ¬ o 9 ∧ ¬ o 11