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

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

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  1. 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

  2. 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

  3. 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

  4. Constraint Hypergraph a b c d e f g h Hypergraph Constraints

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

  6. 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

  7. Observation c b e d a g h i j f c e d 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

  8. The New Backtracking Procedure • Each search-tree node • contains: a subgraph • objective: decompose the subgraph • branching choices: subset of edges

  9. A sample run b d h i j a a c e f g h b j d i a j e f g h i i d a d e a f g b c c a b d h i Branching Choice:

  10. A sample run e c a f g i j a b d h i a c d e c d a b d h i ac d Branching Choice:

  11. A sample run a f g i j c d e c d e c e d a b d h i ac d Branching Choice:

  12. A sample run i c e d a b d h i ac d a f g i gij

  13. No-Good Learning a c d e • 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

  14. Isomorphic subgraphs c d e f g h a b i b d f g h j g a a i e i j a c d f d h i a c d e a g b j g Choice 1 Choice 2

  15. HyperSpread Decomposition a a b d f g h i d g h i a f g b Allow partial branching choices!! • Each HTD is also HSD • HSD is tractable • For some instances hsw<htw • Solves a recently stated problem

  16. Connected Hypertree Decomposition a f g i j i a j c d e f g h b Common variables: a,i a b d h i Restrict choices to edges with: a,i Practically useful variant chtw = htw?

  17. 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/

  18. Methodology d4 d3 d2 k* d1 k*-1 |E| • Limit: 1800 seconds • Test methods: {HTD, CHTD} • ±Isomorphism width ≤|E|? width ≤ k*-1? width ≤ d2? width ≤ d4? k* optimal width

  19. Results overview • We don’t observe htw<chtw NoIso : No Isomorphism

  20. CHTD: Time vs Width Complexity peaks at k*-1 k*:optimal

  21. {CHTD, HTD} ± Isomorphism CHTD much faster than HTD Due to branching restrictions Isomorphism very useful

  22. 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

  23. Thanks !

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