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]