1. Improving Backtrack Search For Solving the TCSP Lin Xu and Berthe Y. Choueiry Constraint Systems Laboratory Department of Computer Science and Engineering University of Nebraska-Lincoln { lxu | choueiry }@cse.unl.edu

2. Outline  Temporal networks  Contributions  Results 2 order of magnitude improvement in solving the TCSP

3. Temporal networks Simple Temporal Problem Floyd-Warshall, Bellman-Ford  STP [Time 03] Disjunctive Temporal Problem Search + heuristics [S&K 00, O&C 00, Tsa&P 03] Some of our results are applicable Temporal Constraint Satisfaction Problem Search + ULT [Schwalb & Dechter 97] Our contribution [this talk]

4. Solving TCSP  TCSP is NP-hard, solved with BT [DM&P 91]  Contributions 1.Combination with previous results  STP [Time 03] 2.Techniques that exploit structure –  AC, a preprocessing step –Show effectiveness of Articulation Points (AP) –NewCyc avoids unnecessary consistency checking –EdgeOrd is a variable ordering heuristic Localized backtracking Implicit decomposition according to Articulation Points (AP) 3.Extensive evaluation on random problems

5. TCSP as a meta-CSP  Use  STP to solve individual STPs efficiently  Especially effective on sparse networks  Requires triangulation: Plan A, Plan B

6. Preprocessing the TCSP  AC Single n-ary constraint GAC is NP-hard   AC Works on existing triangles Poly # of poly constraints

7. Reduction of meta-CSP size

8. 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  It uncovers a phase transition in TCSP

9. New Cycle Check: NewCyc  Check presence of new cycles O(|E|)  Check consistency (  STP) only in a cycle is added to the graph

10. Advantages of NewCyc  Fewer consistency checking operations  Operations restricted to new bi-connected component  Does not affect # of nodes visited in search

11. Edge Ordering in BT-TCSP

12. EdgeOrd heuristic  Order edges using triangle adjacency  Priority list is a by product of triangulation

13. Advantages of EdgeOrd  Localized backtracking  Automatic decomposition of the constraint graph  no need for explicit AP

14. Experimental evaluations  New random generator for TCSPs  Guarantees 80% existence of a solution  Averages over 100 samples  Networks are not triangulated

15. Expected (direct) effects  Number of nodes visited ( #NV )  AC reduces the size of TCSP EdgeOrd localizes BT  Consistency checking effort ( #CC ) AP,  STP, NewCyc, reduce number of consistency checking at each node

16. Effect of  AC on #nodes visited

17. Cumulative improvement 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

18. Future work  Use  AC in a look-ahead strategy  Investigate incremental triangulation for dynamic edge-ordering using NewCyc in Disjunctive Temporal Problem  Plan B, heuristic [G. Noubir], algorithm [A. Berry]  Test with dynamic bundling [AusJCAI 01, SARA 02]