1. IBM Labs in Haifa © 2005 IBM Corporation Assumption-based Pruning in Conditional CSP Felix Geller and Michael Veksler

2. © 2005 IBM Corporation IBM Labs in Haifa 2 Outline  Conditional CSP (Mittal and Falkenhainer)  Conditional CSP solution methods  Activity CSP formulation  MAC-based algorithm for ACSP  Activity disjunction and implication  Questions

3. © 2005 IBM Corporation IBM Labs in Haifa 3  Describes CSP where not all variables are required to participate in a solution  (V, V I, D,C C,C A ) Conditional CSP (Mittal and Falkenhainer)  The assigned variables are active variables  A compatibility constraint is relevant if all its variables are active Activity of a variable is determined by activity constraint Given an assignment of values to  An activity constraint: is relevant if C is relevant  An activity constraint is satisfied if v is active or C is not satisfied  In a solution all relevant constraints have to be satisfied  Minimality requirement

4. © 2005 IBM Corporation IBM Labs in Haifa 4 Existing approaches to solving Conditional CSPs  Reformulation  Working directly on Conditional CSP representation  Main idea: maintain list of active variables and check/propagate constraints after their variables become active  [Mittal,Falkenhainer]: Backtrack search with conflicts recording  [Gelle,Faltings]: At first determine variable activity status; for fixed variable activity status CondCSP turns into CSP.  [Sabin,Freuder,Wallace], [Gelle,Sabin]: invoke consistency techniques (CondFC, CondMAC) directly on Conditional CSP representation  Propagate constraints once their variables are active  CondMAC proved to be superior over CondBT and CondFC  The limitation: constraint propagation has to wait until variables become active

5. © 2005 IBM Corporation IBM Labs in Haifa 5 Activity CSP formulation  (V, V I, V A,D,C,A)  Variable is active if either or is true  Constraint is relevant if all its variables are active  An assignment is a solution if all relevant constraints are satisfied  A - activation conditions: Each variable is dominated by some  Explicit activity variables Given an assignment of values to

6. © 2005 IBM Corporation IBM Labs in Haifa 6  Early conflict detection: e y = true  D(x)= D(y) = {0,…,4} e z = true  D(x)= D(z) = {5,…,9} D(y) ∩ D(z) = Ø  e y  e z = false  Information flow from conditional variables to unconditional variables: e y = true  D(x)= D(y) = {0,…,4} e z = true  D(x)= D(z) = {5,…,9} e y  e z  D(X)= {0…9} An ACSP problem {true,false} eyey ezez {0,…,4} y {5,…,9} z {0,…,100} x = = V

7. © 2005 IBM Corporation IBM Labs in Haifa 7 Enabling early constraint propagation  Allow constraint propagation to modify variable domains before the constraint variables become active eyey ezez {0,…,4} y {5,…,9} z z[e y,e z ] {5,…,9} Activity set ezez eyey y[e y,e z ] {0,…,4} Shadows

8. © 2005 IBM Corporation IBM Labs in Haifa 8 Assumption-based decomposition {0,…4} {5,…,9} {true,false} eyey ezez y z {0,…,100} x = = x eyey ezez y x[e y ] {0,…,4} {0,…,100} = z x[e z ] {5,…,9} {0,…,100} =  Collect assumptions (activity sets)  Create shadows  Group shadows and constraints into CSP sub-problems

9. © 2005 IBM Corporation IBM Labs in Haifa 9 Shadow synchronization y x[e y ] z x[e z ] {0,…,4} {5,…,9} {0,…,100} x eyey ezez {true,false} {0,…,4} {5,…,9} e y  e z = false {3}

10. © 2005 IBM Corporation IBM Labs in Haifa 10 Shadow synchronization  Two shadows of the same variable: x[AS 1 ] and x[AS 2 ]  If under the current partial solution, AS 2 implies AS 1 (AS 1 is weaker than AS 2 )  D(x[AS 2 ])  D(x[AS 2 ]) ∩ D(x[AS 1 ]  If an empty domain results, some activity var in AS 2 is false  If AS 1 and AS 2 are incomparable, but D(x[AS 1 ]) ∩ D(x[AS 2 ]) = Ø  Conclude that some var in AS 1 U AS 2 is false

11. © 2005 IBM Corporation IBM Labs in Haifa 11  Preprocessing  While there are unassigned variables  Select a variable and instantiate it  Propagate constraints  While constraint queue is not empty  Select and propagate a constraint  Synchronize shadows  Update constraint queue  Backtrack if failed AMAC - Putting all things together  Compute constraints activity sets  Create shadows  Substitute constraints parameters with suitable shadows Select an active variable Activity inference  Handle synchronization and constraint propagation failures  Combine activity deductions to determine value of activity variables

12. © 2005 IBM Corporation IBM Labs in Haifa 12 Experimental results - setting  Random Conditional CSP generator similar to the generator in [Sabin,Freuder,Wallace]  CSP over N variables with binary (compatibility) constraints  Problem parameters: Compatibility density Compatibility satisfiability Activity density Activation satisfiability (for )  N=48, |V I |=12, s a =0.75, d c =0.15

13. © 2005 IBM Corporation IBM Labs in Haifa 13 Experimental results – 1 activity cluster

14. © 2005 IBM Corporation IBM Labs in Haifa 14 Experimental results – 4 activity clusters

15. © 2005 IBM Corporation IBM Labs in Haifa 15 Experimental results – 36 independent activation conditions

16. © 2005 IBM Corporation IBM Labs in Haifa 16 Experimental results – effect of clustering (s c =0.25)

17. © 2005 IBM Corporation IBM Labs in Haifa 17 Activity disjunction {0,…,100} y x[e y ] {0,…,4} x z x[e z ] {5,…,9} eyey ezez V {true,false} {12} {0,…,9}

18. © 2005 IBM Corporation IBM Labs in Haifa 18 Activity implication  Implication constraints between activity variables induce equivalence between activity sets  e y  e z  {e y } ~ {e y, e z }  Equivalence allows to reduce the number of shadow variables

19. © 2005 IBM Corporation IBM Labs in Haifa 19 Summary  Introduced Activity CSP – a variant of Conditional CSP formulation  ACSP directly supports full constraint propagation  Activity MAC checks several activity assumptions in parallel  Infers that an assumption is false if the constraint propagation fails  ACSP naturally captures clustering and activity disjunction  Interpret activity disjunction and implication in terms of activity sets and shadows operations  Further improve propagation and information flow  AMAC substantially reduces number of backtracks and solution time on random benchmarks  On easy problems adds overhead of shadow/assumption manipulation

20. IBM Labs in Haifa © 2005 IBM Corporation

21. IBM Labs in Haifa 21 Assumption-based decomposition ConstraintActivity set C 1 (x,e y ) C 2 (x,y) C 3 (y,e z ) C 4 (x,z) C 5 (y,z) eyey ezez y z x eyey eyey ezez ezez eyey x eyey ezez e z [e y ] x[e y ] y z x[e z ] y[e y,e z ] z[e y,e z ]