1. Advanced Constraint Processing, Fall 2009 An Efficient Consistency Algorithm for the Temporal Constraint Satisfaction Problem Berthe Y. Choueiry & Lin Xu, AI Comm. 2004 Presented by Shant Karakashian Advanced Constraint Processing CSCE 990 02, Fall 2009 19/8/2009

2. Advanced Constraint Processing, Fall 2009 Outline Introduction Solving the TCSP Motivation: Constraint Propagation & Fragmentation ∆AC Algorithm Identifying the Triangles ∆AC Initialization Advantages of ∆AC Experiment Results Conclusion 29/8/2009

3. Advanced Constraint Processing, Fall 2009 Introduction Temporal relations in a problem can be represented & reasoned about by modeling the problem as a Temporal Constraint Satisfaction Problem (TCSP) TCSP can be solved by assigning a value to each time point, such that all the constraints are simultaneously satisfied A TCSP can have exponential number of STPs, where each STP can be solved efficiently 39/8/2009

4. Advanced Constraint Processing, Fall 2009 Outline Introduction Solving the TCSP Motivation: Constraint Propagation & Fragmentation ∆AC Algorithm Identifying the Triangles ∆AC Initialization Advantages of ∆AC Experiment Results Conclusion 49/8/2009

5. Advanced Constraint Processing, Fall 2009 Solving TCSP [Dechter et al. 91] Dechter described a backtrack search procedure for determining the consistency of the TCSP. TCSP is modeled as a meta-CSP – With n vertices, e edges, and k intervals per edge – The edges in TCSP are the variables in meta-CSP – The labels of an edge in TCSP form the domain of the corresponding variable – The only constraint is a global constraint, stating that the edge-interval pairs form a consistent STP 59/8/2009

6. Advanced Constraint Processing, Fall 2009 Outline Introduction Solving TCSP Motivation: Constraint Propagation & Fragmentation ∆AC Algorithm Identifying the Triangles ∆AC Initialization Advantages of ∆AC Experiment Results Conclusion 69/8/2009

7. Advanced Constraint Processing, Fall 2009 Constraint Propagation by Domain Filtering Constraint propagation effectively enhances the performance of search The single constraint for the meta-CSP has size O(k e ), which makes it unrealistic to apply GAC ∆AC remedies this difficulty by imposing a new ternary constraint over every existing triangle in the TCSP graph 8/25/2009 Topic 7 Arc consistency –Single n-ary constraint –GAC is NP-hard  AC –Works on existing triangles –Poly # of poly constraints

8. Advanced Constraint Processing, Fall 2009 Fragmentation Algorithms such as NPC-1 and NPC-2 modify intervals, which causes fragmentation  AC removes values that are not supported by the ternary constraint For every interval in the domain of an edge, there must exist intervals in the domains of the 2 other edges such that the 3 intervals verify the triangle inequality rule 8/25/2009 Topic 8 M [1,3] in e 3 has no support in e 1 and e 2  AC removes [1,3] from domain of e 3 ∆AC either keeps an interval as is or removes it, hence avoids fragmentation

9. Advanced Constraint Processing, Fall 2009 Outline Introduction Solving TCSP Motivation: Constraint Propagation & Fragmentation ∆AC Algorithm Identifying the Triangles ∆AC Initialization Advantages of ∆AC Experiment Results Conclusion 99/8/2009

10. Advanced Constraint Processing, Fall 2009 ∆AC Algorithm (1) ∆AC keeps support tables (à la AC-4 / AC-2001) – Calls InitializeSupport Initializes the queue of (edge,interval) pairs Builds 2 support tables: Supports, SupportedBy – Loops through the queue, checking if a (edge-interval) pair has a support in all triangles where the edge appears 8/25/2009 Topic 10 A= {a 1, a 2,.., a k }

11. Advanced Constraint Processing, Fall 2009 ∆AC Algorithm (2) Given the triangle: XYZ R EVISE A={a 1,a, 2,.., a k }, finding a support for each interval For a i in A If F IND S UPPORT (A, a i ) = false Then D ELETE a i from A E NQUEUE the intervals supported by a i F IND S UPPORT (A, a i ) For each interval b j in B, c k in C: If (b j c k )  a i is not empty Then A DD a to list of intervals supported by b j, c k Return true Return false 11 Y XZ C = {c 1,c 2,..,c k } B = {b 1,b 2,..,b k } A= {a 1, a 2,.., a k } 9/8/2009 M [1,3] in e 3 has no support in e 1 and e 2

12. Advanced Constraint Processing, Fall 2009 Outline Introduction Solving TCSP Motivation Filtering intervals with ∆AC Identifying the Triangles ∆AC Initialization Advantages of ∆AC Experiment Results Conclusion 129/8/2009

13. Advanced Constraint Processing, Fall 2009 Identifying the Triangles 13 XY ZW A={a 1, a 2 } C={c 1, c 2 } E={e 1, e 2 } D={d 1, d 2 } F = {[-∞,+∞)} The interval i 11 is an absorbing element: (-∞,+∞) (-∞,+∞) composed with any interval is: (-∞,+∞) (-∞,+∞) intersected with any interval x is x, not null B={b 1, b 2 } 9/8/2009

14. Advanced Constraint Processing, Fall 2009 Outline Introduction Solving TCSP Motivation Filtering intervals with ∆AC Identifying the Triangles ∆AC Initialization Advantages of ∆AC Experiment Results Conclusion 149/8/2009

15. Advanced Constraint Processing, Fall 2009 ∆AC: Initialization 15 (A, a 1 ) (A,a 2 ) (B,b 1 ) (B,b 2 ) (C,c 1 ) (C,c 2 ) (D,d 1 ) (D,d 2 ) Q XY ZW A={a 1, a 2 } C={c 1, c 2 } B={b 1, b 2 } D={d 1, d 2 } d 2 supports b 1 c 1 supports b 1 B={b 1 } C={c 2 } (B,b 1 ) 9/8/2009 b 1 supported by d 2 b 1 supported by c 1

16. Advanced Constraint Processing, Fall 2009 Outline Introduction Solving TCSP Motivation Filtering intervals with ∆AC Identifying the Triangles ∆AC Initialization Advantages of ∆AC Experiment Results Conclusion 169/8/2009

17. Advanced Constraint Processing, Fall 2009 Advantages of  AC Powerful, especially for dense TCSPs Sound and cheap O(n |E| k 3 ) It may be optimal – Uses polynomial-size data-structures: Supports, Supported-by as in AC-4 17 [borrowed from lecture slides] 9/8/2009

18. Advanced Constraint Processing, Fall 2009 Outline Introduction Solving TCSP Motivation Filtering intervals with ∆AC Identifying the Triangles ∆AC Initialization Advantages of ∆AC Experiment Results Conclusion 189/8/2009

19. Advanced Constraint Processing, Fall 2009 Reduction of meta-CSP’s size 19 [borrowed from lecture slides] 9/8/2009

20. Advanced Constraint Processing, Fall 2009 Effect of  AC on #nodes visited 20 [borrowed from lecture slides] 9/8/2009

21. Advanced Constraint Processing, Fall 2009 Cumulative Improvement 21 Before, after AP, after NewCyc,… … and now (  AC,  STP, NewCyc, EdgeOrd) Max on y-axis 5.000.000 Max on y-axis 18.000, 2 orders of magnitude improvement [borrowed from lecture slides] 9/8/2009

22. Advanced Constraint Processing, Fall 2009 Outline Introduction Solving TCSP Motivation Filtering intervals with ∆AC Identifying the Triangles ∆AC Initialization Advantages of ∆AC Experiment Results Conclusion 229/8/2009

23. Advanced Constraint Processing, Fall 2009 Conclusion The paper proposes ∆AC that: – Defines ternary constraints on a TCSP – Propagates the constraints to filter the intervals in the TCSP that are the domains of the variables in the meta-CSP ∆AC is used as a preprocessing step – Should be also used for look-ahead Experiments show that the algorithm is able to filter the domains tremendously specially as the density of the graph increases 239/8/2009