1. An Approximation of Generalized Arc-Consistency for Temporal CSPs Lin Xu and Berthe Y. Choueiry Constraint Systems Laboratory Department of Computer Science and Engineering University of Nebraska-Lincoln { lxu | choueiry }@cse.unl.edu

2. Outline  Temporal CSP  Consistency algorithms For general CSPs: –Arc consistency: AC-1, AC-2, AC-3, AC-4, AC6, AC7, AC2001, AC3.1, …, GAC For Temporal CSPs?   AC

3. STP: example Tom has class at 8:00 a.m. Today, he gets up between 7:30 and 7:40 a.m. He prepares his breakfast (10-15 min). After breakfast (5-10 min), he goes to school by car (20-30 min). Will he be on time for class?

4. Temporal CSP TCSP: each edge is a disjunction of intervals Simple Temporal Problem  Temporal CSP

5. Complexity of consistency  STP is in P Floyd-Warshall algorithm all-pairs shortest path [Dean 85, Dechter et al. 91]  STP some-pairs shortest path [TIME 03]  TCSP is NP-hard Backtrack search [Dechter et al. 91]

6. TCSP as a meta-CSP

7. Filtering by arc-consistency  Arc-consistency Given a constraint, updates the domain of connected variables  AC for TCSP Single n-ary constraint Generalized Arc-Consistency (GAC) is NP-hard

8. Approximating GAC  GAC One global exponential-size constraint   AC Works on existing triangles Polynomial # of polynomial constraints

9.  AC: how it works  Checks combinations of 3 intervals [2, 5] composed with [1, 3] intersects with [3, 6] [1, 3] composed with [3, 6] intersects with [2, 5] M[3, 6] composed with [2, 5] does not intersect with [1, 3]  AC removes [1, 3], not supported, from domain of e 3  Updates the domains of variables, hence  AC  Uses special, polynomial-size data structures Supports, Supported-by

10. Experiments  New random generator for TCSPs  Guarantees 80% existence of a solution  Averages over 100 samples  Networks are not triangulated  Tests demonstrate filtering effectiveness when  AC is used as a preprocessing step Reducing the size of the meta-CSP (i.e., O(k |E| )) Reducing effort for solving the TCSP –Number of constraint checks & CPU time

11. Reduction of meta-CSP size

12. Effect on solving TCSP: CC

13. Effect on solving TCSP: CPU time

14. Advantages of  AC  It is powerful, especially for dense TCSPs  It is sound, effective, and cheap O(n |E| k 3 )  It may be optimal  It uncovers a phase transition in TCSP  Integrated with BT search for TCSP  Last talk at the workshop, today  It should be tested as a look-ahead strategy