Cut-and-Traverse: A new Structural Decomposition Method for CSPs Yaling Zheng and Berthe Y. Choueiry Constraint Systems Laboratory Computer Science & Engineering University of Nebraska-Lincoln yzheng, Tree-Structured DecompositionsCut-and-Traverse (CaT) Decomposition This research is supported by CAREER Award # from the National Science Foundation. General principal 1.Decompose a CSP into sub-problems connected in a tree structure Compute a constraint tree T equivalent to the hypergraph of the CSP Each node in T contains one or more constraints of original CSP 2.Solve each sub-problem (all solutions), usually by a join operation. 3.Apply directional arc-consistency to the constraint tree T. 4.Find a solution for the CSP using backtrack-free search. Goal : a decomposition technique that is efficient and minimizes width of tree 1.Hinge + decomposition: An improvement to hinge decomposition 2.Cut decomposition: A hinge + decomposition bounded by the number of cuts 3.Traverse decomposition: Based on a simple sweep of the constraint hypergraph 4.Cut-and-Traverse decomposition: A combination of the cut and traverse decompositions Given a constraint hypergraph H = (V, E) where H is connected and |E| ≥ k+1. We call a k-cut of H a set F of hyperedges that satisfies the following conditions: 1.F is a subset of E and |F| = k, and 2.The remaining constraint hypergraph H 1, …, H q has at least 2 components. Hinge decomposition of H cg  Hinge decomposition continuously finds 1- cut in in H cg  Width of the 1-hinge tree to the right is 12. Traverse Decomposition We traverse a constraint hypergraph from a set F of hyperedges, until all the hyperedges are visited as follows: Start from F s, mark all hyperedges whose vertices contained in F s as ‘visited.’ Then traverse to F s ’s unvisited neighbors F 1, mark all hyperedges whose vertices contained in F 1 as ‘visited.’ Then traverse to F 1 ’s unvisited neighbors F 2, mark all hyperedges whose vertices contained in F 2 as ‘visited.’ Continuously traversing until all the hyperedges are visited. A traverse decomposition for H cg starting from {s 1 }. Width of the join tree is 3. Future work 1.Empirically evaluate and compare the new proposed decomposition methods on randomly generated constraint hypergraphs. 2.Compare cut-and-traverse decomposition method with hinge decomposition + tree clustering method, and hinge decomposition + biconnected component + hypertree decomposition. References 1.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) 2.Decther, R.: Constraint Processing. Morgan Kaufmann (2003) 3.Freuder, E.C.: A Sufficient Condition for Backtrack-Bounded Search. JACM 32 (4) (1985) 4.Gyssens, M., Jeavons, P.G., Cohen, D.A.: Decomposing Constraint Satisfaction Problems using Database Techniques. Artificial Intelligence 38 (1989) 5.Jeavons, P.G., Cohen, D.A., Gyssens, M. : A structural Decomposition for Hypergraphs. Contemporary Mathematics 178 (1994) 6.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) 8.Harvey, P., Ghose, A.: Reducing Redundancy in the Hypertree Decomposition Scheme. IEEE International Conference on Tools with Artificial Intelligence (ICTAI 03). (2003) 9.Gottlob, G., Leone, N., Scarcello, F.: A comparison of Structural CSP Decomposition Methods. Artifical Intelligence 124 (2000) 10.Gottlob, G., Hutle, M., Wotawa, F.: Combining Hypertree, Bicomp, And Hinge Decomposition. ECAI 02 (2002) 11.Zheng, Y., Choueiry B.Y.: Cut-and-Traverse: A New Structural Decomposition Strategy for Finite Constraint Satisfaction Problems. CSCLP 04 (2004). Cut Decomposition A Cut-and-Traverse decomposition of H cg. Cut limit size is 2. Width of the join tree is 2. A constraint hypergraph H cg.  After finding all the 1-cuts, we continuously find 2-cuts in H cg.  When there are multiple 2-cuts, we choose the one that yields the best division (i.e., the size of the largest sub-problem is the smallest).  Width of the 2-hinge + tree below is 5. Applying hinge decomposition to H cg. Applying hinge + decomposition to H cg. Contribution in Context Hinge + Hypertree [7] TraverseCut-and-Traverse Cut Hinge [4] Tree Clustering   Treewidth [6] Hinge + Tree Clustering [4] Biconnected Component [3] Biconnected Component + Hinge + Hypertree [10] Cut decomposition: A restricted hinge + decomposition. During the process of decomposition, every sub constraint hypergraph contains at least 2 cuts. This constraint tree is not a cut decomposition because the node {s 4, s 5, s 6, s 11, s 12 } contains 3 cuts: {s 4, s 5 }, {s 6, s 12 }, and {s 11 }. Applying cut decomposition to H cg. We traverse a constraint hypergraph from a set F s of hyperedges to another set of hyperedges F d as follows: Start from F s, mark all hyperedges whose vertices contained in F s as ‘visited.’ Then traverse to F s ’s ‘unvisited’ neighbors and those hyperedges in F d that has common vertices with F s, we denote them as F 1, mark all hyperedges whose vertices contained in F 1 as ‘visited.’ Then traverse to F 1 ’s ‘unvisited’ neighbors and those hyperedges in F d that has common vertices with F 1, we denote them as F 2, mark all hyperedges whose vertices contained in F 2 as ‘visited.’ Continuously traversing until traversing to F d and all the hyperedges are visited. A traverse decomposition for H cg starting from {s 1, s 2 } to {s 9, s 16 }. Width of the join tree is 3. Notice that traverse decomposition cannot guarantee a good decomposition result. The result of the decomposition depends on the starting set of hyperedges and ending set of hyperedges. The following graph shows a bad traverse decomposition. A traverse decomposition for H cg starting from {s 6, s 9, s 12 }. Width of the join tree is 10. Cut-and-Traverse decomposition has the following steps: 1. Decompose the constraint hypergraph using cut decomposition. The cut decomposition results in a constraint tree T. 2. For each tree node T, traverse it. If the tree node does not contain any cut, then traverse it from an arbitrary hyperedge. If the tree node contains one cut C 1, then traverse it from C 1. If the tree node contains two cuts C 1 and C 2, then traverse it from C 1 to C Combine the traverse results. Conclusions 2.Hinge + decomposition strongly generalizes hinge decomposition. 3.Cut-and-traverse strongly generalizes cut decomposition. 4.Hypertree decomposition strongly generalizes hinge + decomposition, traverse decomposition, Cut-and-Traverse decomposition. Hinge + decomposition of H cg Hinge + Decomposition Constraint Hypergraph Hinge + decomposition O(|V||E| k+1 ) Cut decomposition O(|V||E| k+1 ) Traverse decompositionO(|V||E| 2 ) Cut-and-Traverse decompositionO(|V||E| k+1 ) k is the limit size for cuts 1.All these decomposition methods can be performed in polynomial time. September 8, 2004