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Structural Decomposition Methods of CSPs

Input: a constraint hypergraph Output: an equivalent join tree. Structural Decomposition Methods of CSPs. s 9. s 2. s 5. s 6. s 3. s 7. s 8. s 10. s 10. 1. 4. 5. 6. 7. 8. 9. 10. 19. s 1. s 5. s 8. s 3. s 7. s 1. s 6. s 17. 0. 2. s 16. s 16. s 9. 18 . 20. s 4.

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Structural Decomposition Methods of CSPs

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  1. Input: a constraint hypergraph Output: an equivalent join tree Structural Decomposition Methods of CSPs s9 s2 s5 s6 s3 s7 s8 s10 s10 1 4 5 6 7 8 9 10 19 s1 s5 s8 s3 s7 s1 s6 s17 0 2 s16 s16 s9 18 20 s4 s11 s13 s2 22 s14 s12 3 11 12 13 14 15 16 17 21 s15 s15 s4 s11 s13 s12 s14 s17 A constraint hypergraph Hcgassociated with a CSP An equivalent join tree of Hcg Criteria to compare structural decomposition methods: (1) CPU time for decomposition (2) Width of the join tree 2004-10-13 Yaling Zheng

  2. Contribution in Context HYPERTREE[1] HYPERCUTSET[11] TRAVERSE CaT HINGE+ HINGETCLUSTER[5] CUT HINGE[5] TCLUSTER[8]  w*[6] = TREEWIDTH[9] BICOMP[3] CUTSET[4] HINGE+: An improvement to HINGE. CUT: A variation of HINGE+. TRAVERSE: Based on a simple sweep of the constraint hypergraph. CaT: A combination of CUT and TRAVERSE. 2004-10-13 Yaling Zheng

  3. i-cut s9 s2 s5 s6 s3 s7 s8 s10 1 4 5 6 7 8 9 10 19 s1 s17 2 s16 0 18 20 22 3 11 12 13 14 15 16 17 21 s15 s4 s11 s13 s12 s14 A 2-cut: {S6, S12}. 2004-10-13 Yaling Zheng

  4. Hinge decomposition (HINGE) HINGE Input: A constraint hypergraph H. Output: An equivalent join tree T of H. The edges of T are labeled. Process: Continuously finds 1-cuts in H. s11 s17 s11 s2 s3 s9 s5 s6 s4 s9 s10 s1 s2 s7 s9 s8 s2 s9 s11 s12 s9 s15 s13 s14 s9 Applying HINGE to Hcg s9 s16 Width = 12 2004-10-13 Yaling Zheng

  5. Hinge+ Decomposition (HINGE+) HINGE+ Input: a constraint hypergraph H and a maximum cut size k. Output: An equivalent join tree T of H. The edges of T are labeled. Process:Finds 1-cuts through k-cuts in H. When there are multiple possible i-cuts, chooses the one that yields the best division (i.e., the size of the largest sub-problem is the smallest). s9 s10 s9 s4 s7 s1 s8 s2 s3 s2 s4 s6 s6 s7 s8 s7 s5 s6 s8 s9 s9 s2 s9 s15 s4 s5 s5 s12 s13 s13 s13 s14 s12 s11 s12 s14 Applying HINGE+ to Hcg, k = 2 s14 s9 s9 s16 s11 s11 s17 Width = 5 2004-10-13 Yaling Zheng

  6. Cut Decomposition (CUT) CUT: A variation of HINGE+. Input: A constraint hypergraph H and a maximum cut size k. Output: An equivalent join tree T of H. The edges of T are labeled. Process:Finds 1-cuts through k-cuts in H. CUT guarantees every tree node of T contains at most 2 different cuts. When there are multiple possible i-cuts that satisfies the condition, choose the one that yields the best division. s9 s10 s9 s3 s6 s6 s1 s8 s3 s2 s3 s5 s5 s6 s8 s2 s6 s7 s4 s7 s7 s9 s9 s9 s15 s2 s4 Applying CUT to Hcg, k = 2 s4 s5 s11 s12 s12 s12 s12 s14 s11 s13 s14 s11 s13 s13 s9 s9 s16 s6 s12 s6 s12 s17 Width = 4 2004-10-13 Yaling Zheng

  7. Traverse Decomposition:TRAVERSE-I TRAVERSE-I Input: A constraint hypergraph H = (V, E) and a set F  E. Output: An equivalent join tree T of H. Process:Sweep through the constraint hypergraph from F. s2 s6 s9 s5 s7 s8 Applying TRAVERSE-I to Hcg, F = {s1} s3 s1 s10 s16 s12 s13 s11 s14 s4 s17 s15 Width = 3 2004-10-13 Yaling Zheng

  8. TRAVERSE-II TRAVERSE-II Input: A constraint hypergraph H = (V, E) and two sets F1, F2 E. Output: An equivalent join tree T of H. Process:Sweep through the constraint hypergraph from F1 to F2. s6 s9 s5 Applying TRAVERSE-II to Hcg, F1 = {s1, s2} F2 = {s9, s16} s1 s3 s7 s8 s9 s10 s12 s13 s11 s4 s14 s16 s2 s17 s15 Width = 3 2004-10-13 Yaling Zheng

  9. Cut-and-Traverse Decomposition (CaT) • CaT: A combination of CUT and TRAVERSE. • Input: A constraint hypergraph H and a maximum cut size k. • Output: An equivalent join tree T of H. • Process:(1) Apply CUT to H and get a join tree Tm • (2) For every tree node Ni inTm , • If it contains no cut, apply TRAVERSE-I to Ni from an arbitrary hyperedge. • If it contains one cut C1, apply TRAVERSE-I to Ni fromC1. • If it contains two cuts C1 and C2, apply TRAVERSE-II from C1 to C2. s10 s6 s5 s3 s7 s8 s2 s12 s1 s13 s9 s16 s11 s4 s14 Applying CaT to Hcg, K = 2. s15 s17 Width = 2 2004-10-13 Yaling Zheng

  10. Preliminary Experiments # Constraints = 20, Maximum arity = 4. For HINGE+, CUT, and CaT, maximum cut size is 2. 2004-10-13 Yaling Zheng

  11. Preliminary Experiments 2004-10-13 Yaling Zheng

  12. All these decomposition methods can be performed in polynomial time. HINGE+O(|V||E|k+1) CUTO(|V||E|k+1) TRAVERSE O(|V||E|2) CaT O(|V||E|k+1) k is the maximum cut size 2. HINGE+ strongly generalizes HINGE. 3. CaT strongly generalizes CUT. 4. HYPERTREE strongly generalizes HINGE+, CUT, TRAVERSE, and CaT. 5. CaT experimentally decomposes better than HINGE, HINGE+, CUT, and TRAVERSE. Conclusions 2004-10-13 Yaling Zheng

  13. More thorough experiments on randomly generated constraint hypergraphs. Compare CaT with HINGETCLUSTER and HINGEHYPERTREE. Future Work 2004-10-13 Yaling Zheng

  14. Gottlob, G., Leone, N., Scarcello, F. : On Tractable Queries and Constraints. In: 10th International Conference and Workshop on Database and Expert System Applications (DEXA 1999). (1999) Decther, R.: Constraint Processing. Morgan Kaufmann (2003) Freuder, E.C.: A Sufficient Condition for Backtrack-Bounded Search. JACM 32 (4) (1985) Dechter, R.: Constraint networks, Encyclopedia of Artificial Intelligence, 2nd edition, Wiley. New York, PP.276-285. (1992) Gyssens, M., Jeavons, P.G., Cohen, D.A.: Decomposing Constraint Satisfaction Problems using Database Techniques. Artificial Intelligence 38 (1989) Jeavons, P.G., Cohen, D.A., Gyssens, M. : A structural Decomposition for Hypergraphs. Contemporary Mathematics 178 (1994) Dechter, R., Pearl. J: Network based heuristic for constraint satisfaction problems, Artificial Intelligence 34 (1) pp 1-38. (1988) Decther, R., Pearl. J: Tree Clustering for Constraint Networks. Artificial Intelligence 38 (1998) Gottlob, G., Leone, N., Scarcello, F.: Hypertree Decompositions and Tractable Queries. Journal of Computer and System Sciences 64. (2002) Robertson, N., Seymour, P.D., Graph Minors II. Algorithmic aspects of tree width, J. Algorithms 7 309-322. (1986) Harvey, P., Ghose, A.: Reducing Redundancy in the Hypertree Decomposition Scheme. IEEE International Conference on Tools with Artificial Intelligence (ICTAI 03). (2003) Gottlob, G., Leone, N., Scarcello, F.: A comparison of Structural CSP Decomposition Methods. Artificial Intelligence 124 (2000) Gottlob, G., Hutle, M., Wotawa, F.: Combining Hypertree, Bicomp, And Hinge Decomposition. ECAI 02 (2002) Zheng, Y., Choueiry B.Y.: Cut-and-Traverse: A New Structural Decomposition Strategy for Finite Constraint Satisfaction Problems. CSCLP 04 (2004). References This research is supported by CAREER Award #0133568 from the National Science Foundation. 2004-10-13 Yaling Zheng

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