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
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
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
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
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