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Structural Decomposition Methods of CSPs • Input: a constraint hypergraph • Output: an equivalent join tree s2 s3 1 4 s1 0 s5 5 s6 6 s7 7 s8 s9 8 9 10 19 s10 s17 18 20 s16 22 3 11 12 13 14 15 16 17 21 s 15 s4 s11 s12 s13 s14 2 A constraint hypergraph Hcg associated with a CSP s1 s2 s3 s4 s5 s6 s11 s12 s10 s7 s13 s8 s9 s14 s16 s15 s17 An equivalent join tree of Hcg Criteria to compare structural decomposition methods: (1) CPU time for decomposition (2) Width of the join tree Constraint Systems Laboratory 2004-10-13 Yaling Zheng 1 Contribution in Context HYPERTREE[1] TRAVERSE CaT CUT HINGE+ HINGETCLUSTER[5] HYPERCUTSET[11] 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. Constraint Systems Laboratory 2004-10-13 Yaling Zheng 2 i-cut s2 s3 1 4 s1 0 s5 5 s6 6 s7 7 s8 s9 8 9 10 19 s10 s17 18 20 s16 22 3 11 12 13 14 15 16 17 21 s 15 s4 s11 s12 s13 s14 2 A 2-cut: {S6, S12}. Constraint Systems Laboratory 2004-10-13 Yaling Zheng 3 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 s11 s17 s1 s 2 s2 Applying HINGE to Hcg s2 s3 s4 s5 s6 s9 s s 9 10 s7 s8 s9 s9 s11 s12 s9 s15 s13 s14 s9 s9 s16 Width = 12 Constraint Systems Laboratory 2004-10-13 Yaling Zheng 4 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). s1 s s s s 2 2 3 4 s2 s4 s5 s5 Applying HINGE+ to Hcg, k=2 s4 s5 s6 s11 s12 s11 s6 s6 s7 s7 s12 s12 s13 s13 s9 s9 s10 s7 s s8 s8 s8 s9 s9 s15 s13 s14 s 9 14 s14 s9 s s 9 16 s11 s17 Width = 5 Constraint Systems Laboratory 2004-10-13 Yaling Zheng 5 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. Applying CUT to Hcg, k=2 s1 s s s s 2 3 3 2 s2 s4 s4 s3 s4 s5 s5 s6 s6 s5 s11 s11 s12 s12 s11 s6 s12 s6 s12 s17 s6 s7 s7 s12 s13 s13 s9 s9 s10 s6 s s7 s8 s8 s9 s s 9 15 s12 s14 s 9 14 s13 s9 s s 9 16 Width = 4 Constraint Systems Laboratory 2004-10-13 Yaling Zheng 6 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. Applying TRAVERSE-I to Hcg, F = {s1} s1 s2 s3 s4 s5 s11 s6 s12 s17 s7 s13 s8 s14 s9 s10 s15 s16 Width = 3 Constraint Systems Laboratory 2004-10-13 Yaling Zheng 7 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. Applying TRAVERSE-II to Hcg, F1 = {s1, s2} F2 = {s9, s16} s1 s2 s3 s4 s5 s11 s6 s12 s17 s7 s13 s8 s14 s9 s10 s15 s9 s16 Width = 3 Constraint Systems Laboratory 2004-10-13 Yaling Zheng 8 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 , a. If it contains no cut, apply TRAVERSE-I to Ni from an arbitrary hyperedge. b. If it contains one cut C1, apply TRAVERSE-I to Ni fromC1. c. If it contains two cuts C1 and C2, apply TRAVERSE-II from C1 to C2. Applying CaT to Hcg, K = 2. s1 s2 s5 s11 s3 s4 s6 s12 s17 s10 s7 s13 s8 s14 s9 s16 s15 Width = 2 Constraint Systems Laboratory 2004-10-13 Yaling Zheng 9 Preliminary Experiments # Constraints = 20, Maximum arity = 4. For HINGE+, CUT, and CaT, maximum cut size is 2. Average CPU time (millseconds) Comparing CPU time 30.00 25.00 20.00 15.00 10.00 5.00 0.00 HINGE HINGE+ CUT CaT TRAVERSE 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Variable Number Constraint Systems Laboratory 2004-10-13 Yaling Zheng 10 Preliminary Experiments Average Width Comparing width 13.00 11.00 9.00 7.00 5.00 3.00 1.00 -1.00 HINGE HINGE+ CUT CaT TRAVERSE 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Variable Number Constraint Systems Laboratory 2004-10-13 Yaling Zheng 11 Conclusions 1. 2. 3. 4. 5. All these decomposition methods can be performed in polynomial time. • HINGE+ O(|V||E|k+1) • CUT O(|V||E|k+1) • TRAVERSE O(|V||E|2) • CaT O(|V||E|k+1) • k is the maximum cut size HINGE+ strongly generalizes HINGE. CaT strongly generalizes CUT. HYPERTREE strongly generalizes HINGE+, CUT, TRAVERSE, and CaT. CaT experimentally decomposes better than HINGE, HINGE+, CUT, and TRAVERSE. Constraint Systems Laboratory 2004-10-13 Yaling Zheng 12 Future Work • • More thorough experiments on randomly generated constraint hypergraphs. Compare CaT with HINGETCLUSTER and HINGEHYPERTREE. Constraint Systems Laboratory 2004-10-13 Yaling Zheng 13 References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 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.276285. (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). This research is supported by CAREER Award #0133568 from the National Science Foundation. Constraint Systems Laboratory 2004-10-13 Yaling Zheng 14