Integrating CSP Decomposition Techniques and BDDs for Compiling Configuration Problems

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Integrating CSP Decomposition Techniques and BDDs
for Compiling Configuration Problems

• Sathiamoorthy Subbarayan
• IT University of Copenhagen
• Denmark
• CP-AI-OR 2005

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Configuration

• An arrangement of parts
• Valid Configuration- a valid arrangement
• Valid according to some specified rules
• NP-Hard problem

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Examples

• Bike
• Car
• PC
• Power-Backup Systems
• Controllers
• Business Software

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Problem Representation

• Constraint Satisfaction Problem (CSP)
• A set of variables and their domains
• A set of rules/constraints on variable values
• Configuration
• - A legal assignment to all variables

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Interactive Configurator

• Interactively guides the user
• Shows only valid extensions
• User always in a consistent state

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Required Functionalities

• Propagate the user selections
• Display the valid choices
• Explain reasons for unselectable or implied
  options

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Binary Decision Diagrams
A compressed representation of solutions
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Previous work

• Compile into a Binary Decision Diagram

configurator
BDD generation
lookup
BDD
BDD
Compilation
rules
offline
online
NP-hard
P
Problem! BDD size could explode
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This work

• Configuration rules have hierarchies
• Hierarchies close to tree-shape
• Tree Decomposition of CSP?
• Space reduction?
• (upto 96)
• Response time reduction?
• (upto 200x)

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CSP to BDD compilation
BDD
CSP
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Tree Decomposition of CSP
CSP
CSP
Tree-of-BDDs
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Benefit of decomposition
BDD
Tree-of-BDDs
Potential Space Reduction
b1b2b3 ?lt bdd
-Online Configurators -Embedded Configurators
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Required Functionalities

• Propagate the user selections
• Display the valid choices
• Explain reasons for unselectable or implied
  options

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Explanations

• A partial assignment by the user might cause
  some options unselectable /implied
• Explanation for such an option is the subset of
  the partial assignment that makes the option
  unselectable/implied
• When user assignments are given priorities, we
  might need minimum explanations

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Explanations in single BDD case

• Variant of Shortest path problem in DAG
• Assigned variable (vval)
• 0 for val edges
• 1 for other edges
• Other variables
• 0 for their edges
• Shortest path from root to 1-node corresponds to
  an explanation

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Example

• Assign a0
• Assign b1
• e1 is implied

1
1
Explain e0?
Explanation b1
Can also handle priorities and non-Boolean
variables
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Dont Care variables for Explain

• In the example, necessary variables are a,b, and
  e
• Variables c and d are hence dont cares for
  Explain

Abstraction !
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Explanations in Tree-of-BDDs
Abstraction
b1
b2
b3
Join
Explanation
Explain
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Experiments

• Four Benchmarks
• PC
• Renault
• Two PSR instances
• 10000 random interactions
• Comparison of
• Space usage
• Average and Worst time for Responses
• Average and Worst time for Explanations
• Tools
• CLab an existing single BDD-based tool
• iCoDE implementation of the tree-of-BDDs

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Compilation Details
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Time Comparisons

• Response (PropagateDisplay)

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Time Comparisons

• Explanation

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Conclusion

• Decomposition results in
• Space reduction
• Shorter responses
• Tree-of-BDDs can scale higher than the single BDD
  approach
• Future work include
• BDD variable orderings
• Multi-valued Decision Diagrams
• different decomposition techniques