1. Bi-intervals for backtracking on temporal constraint networks Jean-François Baget and Sébastien Laborie

2. 2 Motivations My schedule problem (as a temporal constraint network) : Talk Mail Boss Slides meets met-by equals finishes starts meets contains Nodes are time intervals Arcs are disjunctions of Allen’s relations [Allen:83] QUESTION Is there a possible schedule solution ? QUESTION Is there a possible schedule solution ? This problem is NP-complete

3. 3 Related Work Generate all instantiations of relations. Test networks consistency using path-consistency [Allen:83]. Use optimizations: constraints and relations ordering [Van Beek and Manchak:96]… Instantiate constraints : To handle mixed qualitative/quantitative relations : instantiate variables during backtrack To handle mixed qualitative/quantitative relations : instantiate variables during backtrack Use backtracking optimizations such as forward checking [Schwalb et.al.:97] Use backtracking optimizations such as forward checking [Schwalb et.al.:97] Finitely partition the infinite domains of interval values e.g., using an interval end-points encoding [Thornton et.al.:05] Backtrack on these finite partition. Instantiate variables :

4. 4 Outline Instantiate variables with abstract intervals Backtrack with abstract intervals Forward checking with abstract intervals Forward checking (version 2) with abstract intervals Instantiate variables with bi-intervals Conclusion Backtrack with bi-intervals Forward checking with bi-intervals (in the paper)

5. 5 Abstract intervals An abstract interval [x,y] is a pair of elements from a totally ordered list. They encode an infinite number of intervals of real numbers. Definition : Example : Talk Mail meets 17H10 18H30 Talk Mail 17H30 17H25 19H Talk Mail 17H40...... AB C Talk Mail

6. 6 [a,b] possibilities instantiations [a,b] [b,c][d,a] [e,c][a,c][f,c][b,g] XXXX meets met-by equals finishes starts meets contains ab c d [b,c],[d,a] ef g [e,c],[a,c] [f,c],[b,g] Ø Backtrack on abstract intervals Talk Mail Boss Slides talk mail boss mail talk mail boss

7. 7 [a,b] [b,c][d,a] [e,c][a,c][f,c][b,g][h,a][d,i][d,b][d,j] XXXX XX meets met-by a b d [b,c],[d,a] i j [h,a],[d,i] [d,b],[d,j] Ø h [a,b] finishes starts equals meets contains possibilities instantiations Backtrack on abstract intervals Talk Mail Boss Slides talk mail boss slides boss slides

8. 8 [a,b] [b,c][d,a] ab c d [b,c],[d,a] [a,b] [e,c],[a,c] [f,c],[d,c] [g,c],[b,h] e f g h Ø X talk mail talkmail boss slides Forward Checking on abstract intervals [a,b] meets met-by equals finishes starts meets contains Talk Mail Boss Slides possibilities instantiations mail Impossible Slides equals Talk Mail meets Slides

9. 9 slides Slides Boss Forward Checking on abstract intervals [a,b] [b,c][d,a] ab c d [b,c],[d,a] [a,b] [i,a],[d,j] [d,b],[d,k] [d,c],[d,l] j i l X k [d,j][d,b][d,k][d,c][d,l][i,a] [a,b] X X Ø talk mail boss [a,b] meets met-by equals finishes starts meets contains Talk Mail possibilities instantiations

10. 10 Outline Instantiate variables with bi-intervals Instantiate variables with abstract intervals Backtrack with abstract intervals Forward checking with abstract intervals Conclusion Backtrack with bi-intervals Forward checking with bi-intervals

11. 11 Bi-intervals A bi-interval on a list of points encodes a set of abstract intervals [u,v] where u  [A,B] and v  [C,D]. Definition : Example : AB CDE The interpretation of the bi-interval into abstract intervals is : Z X Y [A,C], [A,X], [A,D], [A,Y], [A,E] [Z,C], [Z,X], [Z,D], [Z,Y], [Z,E] [B,C], [B,X], [B,D], [B,Y], [B,E]   [ ] []

12. 12 mailboss Backtrack on bi-intervals talkmail [a,b] [b,c][d,a] ab c d, Ø talk mail meets met-by equals finishes starts meets contains Talk Mail Boss Slides possibilities instantiations --  [e,c][a,c][f,c][b,g] XXXX boss ef g

13. 13 Backtrack on bi-intervals [a,b] [b,c][d,a] abd, Ø talk mail meets met-by equals finishes starts meets contains Talk Mail Boss Slides possibilities instantiations --  slides [h,a][d,i][d,b][d,j] XX [a,b] boss hi j [e,c][a,c][f,c][b,g] XXXX

14. 14 talkmail Forward Checking on bi-intervals [a,b] [b,c][d,a] ab c d, Ø X talk mail boss meets met-by equals finishes starts meets contains Talk Mail Boss Slides possibilities instantiations Impossible Slides equals Talk Mail meets Slides --  slides

15. 15 Forward Checking on bi-intervals [a,b] [b,c][d,a] abd, Ø X talk mail meets met-by equals finishes starts meets contains Talk Mail Boss Slides possibilities instantiations --  slides [e,a][d,f][d,b][d,g] XX [a,b] boss ef g

16. 16 Outline Conclusion Instantiate variables with abstract intervals Backtrack with abstract intervals Forward checking with abstract intervals Instantiate variables with bi-intervals Backtrack with bi-intervals Forward checking with bi-intervals

17. 17 Conclusion Backtrack on abstract intervalsForward checking on abstract intervals Backtrack on bi-intervals Forward checking on bi-intervals Domains are smaller Future Works: Explore other optimizations techniques and use bi-intervals in a hybrid algorithm

18. 18 Thank you for your attention baget@lirmm.fr sebastien.laborie@inrialpes.fr

19. 19 Experimental Results

20. 20 Refining a bi-interval Algorithm: Refining a bi-interval constrained by the finished-by relation. Data: A bi-interval I= and an abstract interval J=[x,y]. Result: Let R be the refinement of I following J and the finished-by relation. We return FALSE if R is empty or a bi-interval whose interpretation in pointsList is R otherwise. if y  (c,d) then return FALSE; if b ≤ x then return FALSE; if a ≥ y then return FALSE; if (a ≤ x and b ≥ y) then return ; if (a ≤ x and b ; if (a > x and b ≥ y) then return ; if (a > x and b ;