Backtracking Procedures for Hypertree, HyperSpread and Connected Hypertree

Decomposition of CSPs

Sathiamoorthy Subbarayan and Henrik Reif Andersen

IT University of CopenhagenDenmark

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

2www.itu.dk

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

a c

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a b g

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c d e

::

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b d h

::

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d e f g

::

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a b

c

d e

f

g

h

Constraints Hypergraph

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Hypertree Decomposition (HTD)

abdhi

acd afgi

gijced

a b d h i

ac d a f g i

gij

a

bc

de

f

gh

i

j

Hypergraph

A Tree Decomposition

A Hypertree Decomposition

Width = 2

c e d

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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

a b d h i

ac d a f g i

gij

a

bc

de

f

gh

i

j

Vars of each rooted subtree form a

connected subgraph

Eg: Root node subtree induces the whole hypergraph

Another example

c e d

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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:

a

b

d

h

i

a

bc

de

f

gh

i

j a

bc

de

f

gh

i

j

a

c

de

af

g

i

j

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A sample run

Branching Choice:

af

g

i

j

a

bc

d

h

i

a

c

de

a b d h i

ac d

c

de

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ac d

A sample run

af

g

i

j

a b d h i

Branching Choice: c

dec

de

c e d

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a f g iac d

A sample run

i

a b d h i

gijc e d

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No-Good Learning

• The next choice needs two edges

• If we need width <2 then we can learn the subgraph as No-Good

a b d h i

a

c

de

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Isomorphic subgraphsa

bc

de

f

gh

i

j

a

b

d

f

gh

i

a

b

d

gh

i

j

Choice 2

Choice 1

a

c

de

g

i

j

a

c

de

af

g

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HyperSpread Decomposition

a

b

d

f

gh

i

a

b

d

gh

i

af

g

Allow partial branching choices!!

•Each HTD is also HSD

•HSD is tractable

•For some instances hsw<htw

•Solves a recently stated problem

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Connected Hypertree Decomposition

a b d h i

af

g

i

j

Common variables: a,i

Restrict choices to edges with: a,i

Practically useful variant

chtw = htw?

a

bc

de

f

gh

i

j

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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-hypertree/

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Methodology

• Limit: 1800 seconds• Test methods: {HTD, CHTD}• ±Isomorphism

|E|k*k*-1 d4 d3 d2 d1

k* optimal width

width ≤|E|? width ≤ d2? width ≤ d4? width ≤ k*-1?

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Results overview

Method Previous tools

CHTD CHTD-NoIso

HTD HTD-NoIso

# optimal at most 1 8 6 5 3

• We don’t observe htw<chtw

NoIso : No Isomorphism

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CHTD: Time vs Width

Complexity peaks at k*-1

k*:optimal

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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 !