An Approximation of Generalized Arc-Consistency for Temporal CSPs Lin Xu and Berthe Y. Choueiry
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Transcript of An Approximation of Generalized Arc-Consistency for Temporal CSPs Lin Xu and Berthe Y. Choueiry
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