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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
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Outline Temporal networks Contributions Results 2 order of magnitude improvement in solving the TCSP
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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]
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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
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TCSP as a meta-CSP Use STP to solve individual STPs efficiently Especially effective on sparse networks Requires triangulation: Plan A, Plan B
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Preprocessing the TCSP AC Single n-ary constraint GAC is NP-hard AC Works on existing triangles Poly # of poly constraints
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Reduction of meta-CSP size
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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
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New Cycle Check: NewCyc Check presence of new cycles O(|E|) Check consistency ( STP) only in a cycle is added to the graph
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Advantages of NewCyc Fewer consistency checking operations Operations restricted to new bi-connected component Does not affect # of nodes visited in search
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Edge Ordering in BT-TCSP
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EdgeOrd heuristic Order edges using triangle adjacency Priority list is a by product of triangulation
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Advantages of EdgeOrd Localized backtracking Automatic decomposition of the constraint graph no need for explicit AP
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Experimental evaluations New random generator for TCSPs Guarantees 80% existence of a solution Averages over 100 samples Networks are not triangulated
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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
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Effect of AC on #nodes visited
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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
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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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