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Improving Backtrack Search For Solving the TCSP Lin Xu and Berthe Y. Choueiry

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. Outline. Temporal networks Contributions Results

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Improving Backtrack Search For Solving the TCSP Lin Xu and Berthe Y. Choueiry

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  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] • Temporal Constraint Satisfaction Problem • Search + ULT [Schwalb & Dechter 97] • Our contribution [this talk] • Disjunctive Temporal Problem • Search + heuristics [S&K 00, O&C 00, Tsa&P 03] • Some of our results are applicable

  4. Solving TCSP • TCSP is NP-hard, solved with BT [DM&P 91] • Contributions • Combination with previous results STP [Time 03] • 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) • 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. AC Single n-ary constraint GAC is NP-hard AC Works on existing triangles Poly # of poly constraints Preprocessing the TCSP

  7. Reduction of meta-CSP size

  8. Advantages of AC • Powerful, especially for dense TCSPs • Sound and cheap O(n |E| k3) • 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 18.000, 2 orders of magnitude improvement Max on y-axis 5.000.000

  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]

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