Efficient Techniques for Searching the Temporal CSP Lin Xu and Berthe Y. Choueiry Constraint Systems Laboratory Department of Computer Science and Engineering.

Efficient Techniques for Searching the Temporal CSP

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

• 2 order of magnitude improvement on TCSP

Temporal networksSimple Temporal Problem• Floyd-Warshall algorithm [Dean 85, Dechter et al. 91]

• 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, CP 03]

Solving TCSP TCSP is NP-hard, solved with BT [DM&P 91]

Contributions1. Techniques that exploit structure

– 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)

2. Combination with previous results – AC, a preprocessing step [this morning]

– STP [Time 03]

3. Extensive evaluation on random problems

TCSP as a meta-CSP

• Preprocessing with AC reduces size of TCSP, especially for dense networks• Using STP solves individual STPs efficiently, especially for sparse networks

requires triangulation: Plan A, Plan B

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 evaluationsWith/without: Explicit decomposition using AP, AC, STP, NewCyc, EdgeOrd

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

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]