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

Outline  Temporal networks  Contributions  Results 2 order of magnitude improvement on TCSP

Temporal networks Simple 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]  Contributions 1.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 evaluations With/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 improvement Before, after AP, after NewCyc,… … and now (  AC,  STP, NewCyc, EdgeOrd) Max on y-axis Max on y-axis , 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]