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]