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]