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

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

STP: exampleTom 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?

Temporal CSP

TCSP: each edge is a disjunction of intervals

Simple Temporal Problem Temporal CSP

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]

TCSP as a meta-CSP

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

Approximating GAC GAC

• One global exponential-size constraint

AC

• Works on existing triangles• Polynomial # of polynomial constraints

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] [3, 6] composed with [2, 5] does not intersect with [1, 3]

AC removes [1, 3], not supported, from domain of e3

Updates the domains of variables, hence AC Uses special, polynomial-size data structures

• Supports, Supported-by

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

Reduction of meta-CSP size

Effect on solving TCSP: CC

Effect on solving TCSP: CPU time

Advantages of AC

It is powerful, especially for dense TCSPs

It is sound, effective, and cheap O(n |E| k3)

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