IndSet: A Decomposition Technique for CSPs

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• IndSet A Decomposition Technique for CSPs
• Using Maximal Independent Sets
• Its Integration with Local Search
• Reduces the time to solve a CSP
• Returns multiple solutions
• Joel Gompert and Berthe Y. Choueiry
• Constraint Systems Laboratory
• Department of Computer Science Engineering
• University of Nebraska-Lincoln

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Main contributions

• IndSet, a new structural decomposition technique
• Combination with local search (LS)
• Heuristics for integration with LS
• Experiments showing
• Many solutions found
• Reduction in runtime

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Robust solutions

• Single solution
• V1 d
• V2 e
• V3 a
• V4 c

• Robust solution
• V1 d
• V2 d, e, f
• V3 a
• V4 b, c

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IndSet decompose
Decompose CSP into I I
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IndSet solve
Solve I, using any technique
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IndSet propagate
Apply DAC Revise(I, I)

• If no domain in I is empty, we have
• solved the original CSP
• found multiple solutions (cross product of
  domains in I)

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SLS/IndSet 5 heuristics

• All

• None

• Some

• Zero-domain

• PrefRelax Junker, 04

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SLS/IndSet Results
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SLS/IndSet Results
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Finding dangles

• Identify T dangles
• Perform DAC T?I
• Apply SLS/IndSet on C I instead of on I
  I
• Extend to T, in parallel

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Effect of dangles

• Reduces the size of the cutset
• Increases the number of solutions
• Slightly improves runtime

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Number of Solutions Found
2
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Log of number of solutions
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Runtime Evaluation
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Cycle-cutset vs. IndSet

• Cycle-Cutset leaves the graph with no cycles

DanglesIndSet
IndSet leaves the graph with no edges
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Related work decompositions

• IndSet is
• A conjunctive decomposition Freuder, 95
• A special case of Cycle-Cutset Dechter Pearl,
  87
• A backdoor variables technique Williams et al.,
  03
• Produces multiple solutions Choueiry et al.,
  95
• Similar to dependent variables in SAT Kautz et
  al., 97

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Future work (more in dissertation)

• Perform experiments on more problems
• real-world problems
• k-regular graphs
• Explore how to compactly represent all solutions
  in a tree graph
• Explore how to increase the robustness of
  solutions
• Investigate how to solve k-partite graphs

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• 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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• Additional slides

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Outline

• Background
• IndSet
• Explorations
• NI clustered graphs
• Runtime variance
• Finding dangles first
• Recursive decomposition
• Using IndSet to find cycle-cutsets
• (and more in thesis)
• Related work future research

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Run-time variance
Trial 1 Trial 2 Trial 3 Trial 4
Instance 1 8.74 9.02 4.77 38.71
Instance 2 27.01 27.89 4.97 4.63
Instance 3 8.99 8.98 4.48 8.70
Instance 4 4.10 53.60 17.16 8.72

SLS SLS/IndSetDangles
Row variance 67.83 24.02
Column variance 68.45 24.12
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Finding dangles first
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Recursive decomposition
A
Ia
B
I
I
Ib
C
Ic

• RecIndSet
• RecCliq repeatedly find remove cliques

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Using IndSet to find cycle cutsets
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Outline

• Background
• IndSet
• Explorations
• Related work future research

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Effect of size of I

• Tightness 60
• Constraint ratio 3.3

Size of I Runtime (seconds) SLS / IndSet
0 147.20
32 4.86
41 1.64
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Independent sets

Finding maximum independent sets is NP-hard

• We use polynomial-time CliqueRemoval Boppana
  Halldórsson, 90

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Background CSP

• Constraint Satisfaction Problem CSP
• P (V, D, C)
• Given
• V V1, V2, V3, , VN
• D Dv1, Dv2, Dv3, , DN
• C set of relations on subsets of D
• Question Can we assign a value to each variable
  such that all constraints are satisfied?
• (i.e., find a consistent solution)

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CSP characteristics

• NP-Complete
• Binary CSP
• Constraint graph
• Constraint ratio
• Tightness