October 2nd, 2005 Zheng – DocProg CP’05 1

Constraint Systems Laboratory

Applying Structural Decomposition Methods to Crossword P

uzzle Problems

Student: Yaling ZhengAdvisor: Berthe Y. Choueiry

Constraint Systems LaboratoryDepartment of Computer Science & Engineering

University of Nebraska-Lincoln

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Constraint Systems Laboratory

HINGETCLUSTER

Gyssens et al., 1994HYPERCUTSETGottlob et al., 2000

TCLUSTERDechter & Pearl,

1989

BICOMPFreuder, 1985

HYPERTREEGottlob et al., 2002

HINGEGyssens et al.,

1994

CaT

CUT

HINGE+ TRAVERSE

CUTSETDechter, 1987

Structural decomposition methods

Criteria for comparing decomposition methods:1. Width of an x-decomposition = largest number of hyperedges i

n a node of the tree generated by x-decomposition2. CPU time for generating the tree

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Constraint Systems Laboratory

S1 S2

S4

S3 S5 S7S6

S15

S8

S12 S13 S14S11

S10

S9

S16S17

Constraint hypergraph

• A vertex represents a variable• A hyperedge represents a constraint (delimits it

s scope)• Cut is a set of hyperedges whose removal discon

nects the graph• Cut size is the number of hyperedges in the cut

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Constraint Systems Laboratory

S2

S4

S3 S5 S7S6 S8

S12 S13 S14S11

S9

width = 12

S1 S2

S4

S3 S5 S7S6

S15

S8

S12 S13 S14S11

S10

S9

S16S17

S10S9

S16S9

S15S9

S17S11

S1 S2

In HINGE, the cut size is limited to 1

HINGE [Gyssens et al., 94]

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Constraint Systems Laboratory

HINGE+: maximum cut size is a parameter

S2

S4

S3

S5

S7

S6

S12

S13

S11S9

S16

S17

S9

S11

S5

S4S6

S12

S8

S14

width = 5

S1 S2

S4

S3 S5 S7S6

S15

S8

S12 S13 S14S11

S10

S9

S16S17

S8

S14

S7

S13

S10S9

S15S9

S1 S2

HINGE+ with maximum cut size of 2:

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Constraint Systems Laboratory

CUT

S2

S4

S3 S7

S8

S13

S14

S11S9

S17

S5

S6

S12

S7

S13

S8

S14

S6 S12

S4

S3

S11

S5

S6

S12

S1 S2

S4

S3 S5 S7S6

S15

S8

S12 S13 S14S11

S10

S9

S16S17

width = 4

S10S9

S16S9

S15S9

S1 S2

A tree node contains at most 2 of the selected cuts

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Constraint Systems Laboratory

TRAVERSE-I

S1

S2

S3

S4

S5

S11

S6

S12

S17

S7

S13

S8

S14

S9

S10

S15

S16

S1 S2

S4

S3 S5 S7S6

S15

S8

S12 S13 S14S11

S10

S9

S16S17

width = 3

• TRAVERSE-I starts from one set of hyperedges, and• Sweeps through the network

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Constraint Systems Laboratory

TRAVERSE-II

S2

S3

S4

S5

S11

S6

S12

S17

S7

S13

S8

S14

S9

S10

S15

S9

S16

S2

S4

S3 S5 S7S6

S15

S8

S12 S13 S14S11

S10

S9

S16S17

width = 3

• TRAVERSE-II starts from one set of hyperedges {s1},• Sweeps the network, and • Ends at another set of hyperedges {s9, s16}.

S1

S1

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Constraint Systems Laboratory

CaT: combines CUT and TRAVERSE

S1

S2

S3

S4

S7

S8

S13

S14

S11

S10

S9

S17

S2

S9S5

S6

S12

S7

S13

S6 S12

S11

S5

CUT

S1S2

S4

S3 S5 S7S6

S15

S8

S12 S13 S14S11

S10

S9

S16

S17

S4

S3S6

S12

S8

S14

S16S9

S15S9

S1S7 S8

S13 S14S11

S10

S9

S17

S5

S4

S3 S6

S12S16

S15

S2

width = 2

Original

CaT

width = 4

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Constraint Systems Laboratory

Experiments• Test problems:

– Random CSPs– 51 instances of fully interlocked CPPs, from C

rossword Puzzle Grid Library puzzles.about.com/library

• Compared: – HINGE, HINGE+, CUT, TRAVERSE, and CaT – (Hypertree was too expensive to run on CPPs)

• Criteria: width and CPU time

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Constraint Systems Laboratory

Results on random CSPs

• CPU time:

• Width:

TRAVERSE CUT CaT HINGE+ HYPERTREE

TRAVERSEHYPERTREE CaT HINGE+ CUT HINGE

HINGE

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Constraint Systems Laboratory

CPPs: CPU time (ms)

0

100000

200000

300000

400000

500000

600000

700000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 HINGE

HINGE+

CaT

TRAVERSE

Instance ID

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Constraint Systems Laboratory

CPPs: width

HINGE

HINGE+

CaT

TRAVERSE

Instance ID

0

20

40

60

80

100

120

140

160

180

200

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

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Constraint Systems Laboratory

CPP: Notable exception

• On only 1 CPP, HINGE outperforms CaT

HINGE CaT

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Constraint Systems Laboratory

Current investigations

• Test on special constraint hypergraphs • Study the cost of solving the CSPs after

decomposition [Jégou and Tierroux, 03]

• Preprocessing by ‘subproblem’ elimination– Recursively eliminate vertices that appear in

one hyperedge– Solve and eliminate subproblems that have f

ew solutions (tight constraints)

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Constraint Systems Laboratory

Thank you for your attention.

Research supported by CAREER Award #0133568 from NSF. Experiments conducted on PrairieFire of the Research Computing Facilities (RCF) of CSE-UNL.

Questions…

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Constraint Systems Laboratory

Additional Slides

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Constraint Systems Laboratory

How do we choose a cut?

S2

S4

S3 S5 S7S6 S8

S12 S13 S14S11

S9

S17

S2

S4

S3 S5 S7S6 S8

S12 S13 S14S11

S9

S17

When there are multiple cuts to choose, we choose the cut whose removal makesthe greatest number of hyperedges in a connected hypergraph the smallest.

8

7

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Constraint Systems Laboratory

TRAVERSE

When applying TRAVERSE to these CPPs, we choose the decomposition whose width is the smallest among all the decompositions that starts from an arbitrary hyperedge.