Improving Backtrack Search For Solving the TCSP Lin Xu and Berthe Y. Choueiry
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Transcript of Improving Backtrack Search For Solving the TCSP Lin Xu and Berthe Y. Choueiry
Improving Backtrack Search For Solving the TCSPLin Xu and Berthe Y. Choueiry
Constraint Systems LaboratoryDepartment of Computer Science and Engineering
University of Nebraska-Lincoln{ lxu | choueiry }@cse.unl.edu
Outline
Temporal networks
Contributions
Results• 2 order of magnitude improvement in
solving the TCSP
Temporal networksSimple 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]
Solving TCSP TCSP is NP-hard, solved with BT [DM&P 91]
Contributions1. 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
TCSP as a meta-CSP
Use STP to solve individual STPs efficiently Especially effective on sparse networks Requires triangulation: Plan A, Plan B
Preprocessing the TCSP
AC• Single n-ary constraint• GAC is NP-hard
AC• Works on existing triangles• Poly # of poly constraints
Reduction of meta-CSP size
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
New Cycle Check: NewCyc
Check presence of new cycles O(|E|) Check consistency (STP) only in a cycle is
added to the graph
Advantages of NewCyc Fewer consistency checking operations Operations restricted to new bi-connected
component
Does not affect # of nodes visited in search
Edge Ordering in BT-TCSP
EdgeOrd heuristic Order edges using triangle adjacency Priority list is a by product of triangulation
Advantages of EdgeOrd Localized backtracking Automatic decomposition of the constraint graph
no need for explicit AP
Experimental evaluations
New random generator for TCSPs Guarantees 80% existence of a solution Averages over 100 samples Networks are not triangulated
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
Effect of AC on #nodes visited
Cumulative improvementBefore, 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
Future work
Use AC in a look-ahead strategy Investigate incremental triangulation for
• dynamic edge-ordering
• using NewCyc in Disjunctive Temporal ProblemPlan B, heuristic [G. Noubir], algorithm [A. Berry]
Test with dynamic bundling [AusJCAI 01, SARA 02]