1. Advanced Constraint Processing, Fall 2009 On the Relationship Between Partial Interchangeability and Subproblem Interchangeability Shant Karakashian & Christopher Reeson Advanced Constraint Processing CSCE 990 02, Fall 2009 19/8/2009

2. Advanced Constraint Processing, Fall 2009 Outline Definitions – Partial interchangeability (PI) – Subproblem interchangeability (SubPI) Relationship between PI & SubPI – Theorem 5 of [Freuder, AAAI91] – Examples 29/8/2009

3. Advanced Constraint Processing, Fall 2009 Definitions Two values are partially interchangeable with respect to a subset S of variables iff any solution involving one implies a solution involving the other with possibly different values for S. Two values are subproblem interchangeable with respect to a subset of variables S iff they are fully interchangeable with respect to the solutions of the subproblem of the CSP induced by S. 8/25/2009 Topic 3

4. Advanced Constraint Processing, Fall 2009 Theorem 5 of [Freuder AAAI91] Subproblem interchangeability wrt S implies partial interchangeability wrt S’=V\S SI  PI Partial interchangeability wrt S’ does not imply subproblem interchangeability wrt S. PI  (not  ) SI Examples: one for PI   SI; one for  PI  SI 8/25/2009 Topic 4

5. Advanced Constraint Processing, Fall 2009 Example: PI   SI 8/25/2009 Topic 5 AB 00 01 02 11 12 BC 10 11 20 21 AC 00 11 ABC 010 020 111 121 AB 00 01 02 11 12 0101 012012 0101 0101 012012 0101 012012 0101 CSP A B AA C B BC Values 0,1 for A are not SI in {A,B} Values 0,1 for A are PI wrt {C}

6. Advanced Constraint Processing, Fall 2009 Example:  PI  SI 8/25/2009 Topic 6 AB 00 01 10 11 BC 10 AC 00 ABC 010 AB 00 01 10 11 0101 0101 0 CSP A B C Values 0,1 for A are SI in {A,B} Values 0,1 for A are not PI wrt {C} 0101 0101 A B 0101 0101 0 A B