Backtracking Procedures for Hypertree, HyperSpread and Connected Hypertree Decomposition of CSPs

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Backtracking Procedures for Hypertree,
HyperSpread and Connected HypertreeDecomposition
of CSPs

• Sathiamoorthy Subbarayan and Henrik Reif Andersen
• IT University of Copenhagen
• Denmark

Twentieth International Joint Conference on
Artificial Intelligence 06 12 Jan 2007,
Hyderabad, India
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Motivation

• Many CSPs have tree-like structure
• Configuration, fault trees, digital circuits,
  protein side-chain packing, Bayesian networks
  etc.,
• Hypertree decomposition the most general in
  theory, lacks practical tools
• Very weak existing tools

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This Work

• Backtracking for hypertree decomp. (HTD)
• No-goods and Isomorphism
• New Tractable Variants
• HyperSpread (HSD), Connected Hypertree (CHTD)
• HSD is better than HTD
• solves a recent problem
• CHTD htw chtw?
• Experiments

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Constraint Hypergraph
Hypergraph
Constraints
a c

a b g

b d h

c d e

d e f g

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Hypertree Decomposition (HTD)
Hypergraph
A Hypertree Decomposition
a b d h i
ac d
a f g i
A Tree Decomposition
abdhi
gij
c e d
acd
afgi
Width 2
gij
ced
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Hypertree Width (htw)

• Complexity exponential in htw
• Advantage more general than treewidth (tw)
• For any class H htw is bounded by tw
• For some class H unbounded tw, constant htw

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Observation
Vars of each rooted subtree form a connected
subgraph
Eg Root node subtree induces the whole hypergraph
Another example
a b d h i
ac d
a f g i
gij
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The New Backtracking Procedure

• Each search-tree node
• contains a subgraph
• objective decompose the subgraph
• branching choices subset of edges

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A sample run
a b d h i
Branching Choice
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A sample run
a b d h i
ac d
Branching Choice
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A sample run
a b d h i
ac d
Branching Choice
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A sample run
a b d h i
ac d
a f g i
gij
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No-Good Learning

• The next choice needs two edges
• If we need width lt2 then we can learn the
  subgraph as No-Good

a b d h i
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Isomorphic subgraphs
Choice 1
Choice 2
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HyperSpread Decomposition
Allow partial branching choices!!

• Each HTD is also HSD
• HSD is tractable
• For some instances hswlthtw
• Solves a recently stated problem

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Connected Hypertree Decomposition
Common variables a,i
a b d h i
Restrict choices to edges with a,i
Practically useful variant
chtw htw?
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Experiments

• Intel Xeon 3.2 GHz, 4GB RAM
• Twelve instances configuration, fault trees, SMT
• Tools and instances online
• http//www.itu.dk/people/sathi/connected-hypert
  ree/

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Methodology

• Limit 1800 seconds
• Test methods HTD, CHTD
• Isomorphism

width E?
width d2?
width d4?
width k-1?
k optimal width
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Results overview
Method Previous tools CHTD CHTD-NoIso HTD HTD-NoIso
optimal at most 1 8 6 5 3
NoIso No Isomorphism

• We dont observe htwltchtw

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CHTD Time vs Width
Complexity peaks at k-1 koptimal
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CHTD, HTD Isomorphism
CHTD much faster than HTD Due to branching
restrictions
Isomorphism very useful
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Conclusion

• Backtracking useful
• Isomorphism and No-good
• HSD better than HTD
• CHTD promising for practice
• Future work
• htw chtw ?
• implement HSD
• compare tree decomposition heuristics

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Thanks !