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```									                                    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, choueiry@cse.unl.edu
Tree-Structured Decompositions                                                                            Cut Decomposition                                                                       Cut-and-Traverse (CaT) Decomposition
General principal                                                                              Cut decomposition: A restricted hinge+ decomposition. During the process of                           Cut-and-Traverse decomposition has the following steps:
1. Decompose a CSP into sub-problems connected in a tree structure                             decomposition, every sub constraint hypergraph contains at least 2 cuts.                              1. Decompose the constraint hypergraph using cut decomposition.
• Compute a constraint tree T equivalent to the hypergraph of the CSP                                                                                                                                 The cut decomposition results in a constraint tree T.
• Each node in T contains one or more constraints of original CSP                                             s4        s6                                    This constraint tree is not         2. For each tree node T, traverse it.
s2         s5 s6              s7        s9 s9 s10
2. Solve each sub-problem (all solutions), usually by a join operation.
s1 s2 s3      s4
s11
s6    s7 s7 s8 s8 s8 s                      a cut decomposition because             If the tree node does not contain any cut, then traverse it from an arbitrary
3. Apply directional arc-consistency to the constraint tree T.
s2    s4      s5 s s12      s12 s s s s9 9 s9 s15                  the node {s4, s5, s6, s11, s12}          hyperedge.
13 13 14 s
4. Find a solution for the CSP using backtrack-free search.
s5            12      s13    s14      14 s9                  contains 3 cuts:                        If the tree node contains one cut C1, then traverse it from C1.
Goal: a decomposition technique that is efficient and minimizes width of tree                                                                   s s    9 16        {s4, s5}, {s6, s12}, and {s11}.        If the tree node contains two cuts C1 and C2, then traverse it from C1 to C2.
s11                                                                                       

3. Combine the traverse results.
Contribution in Context                                                            s11 s17
s9s10
s10
Hypertree     [7]                                                          s3     s5        s6         s7          s9                                                  s1 s2        s3    s5 s6 s7              s8 s                   A Cut-and-Traverse
s1 s s2 s s s s s                                s8                                                                        s11 s12 s13
2     3 4 5 6       6    s7 s7     s8 s8     s9 s9 s9 s15       Applying                                         s4                          s14  9 s
15
decomposition of Hcg.
Hinge                                      s2    s3 s s s s             s12 s13   s13 s14                      cut decomposition to Hcg.                                                                           Cut limit size is 2.
Biconnected Component
s4 4 s 5 11 s 11 s12                       s14 s9
Traverse Cut-and-Traverse        Hinge+              +                                                                     s13       s14                                                                                                             s16              Width of the join tree is 2.
+ Hinge                          11     12                                 s9 s16                                                                     s17
Tree Clustering [4]     + Hypertree [10]                         s6 s12
Cut                                                                                       s6 s12 s17
Hinge [4]         Tree Clustering 
 Treewidth [6]                                                                                                                                                                        Conclusions
Biconnected Component [3]                                                                                                                      1. All these decomposition methods can be performed in polynomial time.
Traverse Decomposition
Hinge+ decomposition O(|V||E|k+1)
1. Hinge+ decomposition: An improvement to hinge decomposition
We traverse a constraint hypergraph from a set F of hyperedges, until all the                                    Cut decomposition O(|V||E|k+1)
2. Cut decomposition: A hinge+ decomposition bounded by the number of cuts
hyperedges are visited as follows:                                                                              Traverse decompositionO(|V||E|2)
3. Traverse decomposition: Based on a simple sweep of the constraint hypergraph
   Start from Fs, mark all hyperedges whose vertices contained in Fs as ‘visited.’                             Cut-and-Traverse decompositionO(|V||E|k+1)
4. Cut-and-Traverse decomposition: A combination of the cut and traverse
   Then traverse to Fs’s unvisited neighbors F1, mark all hyperedges whose                                     k is the limit size for cuts
decompositions
vertices contained in F1 as ‘visited.’                                                            2. Hinge+ decomposition strongly generalizes hinge decomposition.
   Then traverse to F1’s unvisited neighbors F2, mark all hyperedges whose
Constraint Hypergraph                                                     vertices contained in F2 as ‘visited.’
3. Cut-and-traverse strongly generalizes cut decomposition.
4. Hypertree decomposition strongly generalizes hinge+ decomposition, traverse
s2 s3  s5     s6 s7 s8 s9                                      Continuously traversing until all the hyperedges are visited.                                        decomposition, Cut-and-Traverse decomposition.
s10
1 4 5 6 7 8 9 10 19                                                    s2     s5 s6 s7                     s9
s1                                 s8                       A traverse decomposition for Hcg
s1 0 2                                                                       s3     s11 s12 s13                  s10     s16                                                                                Future work
A constraint hypergraph Hcg .            s17 22         18 20 s16                                                                     s14
s15
starting from {s1}.
s4         s17                                      Width of the join tree is 3.
3 11 12 13 14 1516 17 21 s15                                                                                                                                     1. Empirically evaluate and compare the new proposed decomposition methods
s4 s11 s12 s13 s14                                                                                                                                                 on randomly generated constraint hypergraphs.
We traverse a constraint hypergraph from a set Fs of hyperedges to another set of
2. Compare cut-and-traverse decomposition method with
hyperedges Fd as follows:
• hinge decomposition + tree clustering method, and
Given a constraint hypergraph H = (V, E) where H is connected and |E| ≥ k+1. We                  Start from Fs, mark all hyperedges whose vertices contained in Fs as ‘visited.’
• hinge decomposition + biconnected component + hypertree decomposition.
call a k-cut of H a set F of hyperedges that satisfies the following conditions:              Then traverse to Fs’s ‘unvisited’ neighbors and those hyperedges in Fd that has
1. F is a subset of E and |F| = k, and                                                            common vertices with Fs, we denote them as F1, mark all hyperedges whose
2. The remaining constraint hypergraph H1, …, Hq has at least 2 components.                       vertices contained in F1 as ‘visited.’
   Then traverse to F1’s ‘unvisited’ neighbors and those hyperedges in Fd that has                                                          References
common vertices with F1, we denote them as F2, mark all hyperedges whose                          1.  Gottlob, G., Leone, N., Scarcello, F. : On Tractable Queries and Constraints. In: 10th
Hinge+ Decomposition                                                   vertices contained in F2 as ‘visited.’                                                                International Conference and Workshop on Database and Expert System Applications
s2 s3 s11 s11s17        Continuously traversing until traversing to Fd and all the hyperedges are visited.                    (DEXA 1999). (1999)
2. Decther, R.: Constraint Processing. Morgan Kaufmann (2003)
Hinge decomposition of Hcg                                  s1 s2      s4 s5 s s9
 Hinge decomposition continuously finds 1-
6    s9 s10    s1      s3        s5 s6 s7        s8       s9          s9    A traverse decomposition for Hcg         3. Freuder, E.C.: A Sufficient Condition for Backtrack-Bounded Search. JACM 32 (4) (1985)
s2         s7 s8 s9 s9             s2      s4        s11 s12 s13              s10               starting from {s1, s2} to {s9, s16}.     4. Gyssens, M., Jeavons, P.G., Cohen, D.A.: Decomposing Constraint Satisfaction
cut in in Hcg                                                         s11 s12      s9 s15                                      s14                  s16                                                Problems using Database Techniques. Artificial Intelligence 38 (1989)
s17                  s15               Width of the join tree is 3.
 Width of the 1-hinge tree to the right is 12.                         s13 s14 s9                                                                                                                   5. Jeavons, P.G., Cohen, D.A., Gyssens, M. : A structural Decomposition for Hypergraphs.
s9 s16                                                                                                               Contemporary Mathematics 178 (1994)
6. Decther, R., Pearl. J: Tree Clustering for Constraint Networks. Artificial Intelligence 38
Applying hinge decomposition to Hcg.                                                                                                               (1998) Gottlob, G., Leone, N., Scarcello, F.: Hypertree Decompositions and Tractable
Notice that traverse decomposition cannot guarantee a good decomposition result.
Hinge+ decomposition of Hcg                                                                   The result of the decomposition depends on the starting set of hyperedges and ending                       Queries. Journal of Computer and System Sciences 64 (2002)
set of hyperedges. The following graph shows a bad traverse decomposition.                             8. Harvey, P., Ghose, A.: Reducing Redundancy in the Hypertree Decomposition Scheme.
 After finding all the 1-cuts, we continuously find 2-cuts in Hcg.
IEEE International Conference on Tools with Artificial Intelligence (ICTAI 03). (2003)
 When there are multiple 2-cuts, we choose the one that yields the best division (i.e.,                                                                                                             9. Gottlob, G., Leone, N., Scarcello, F.: A comparison of Structural CSP Decomposition
the size of the largest sub-problem is the smallest).
 Width of the 2-hinge+ tree below is 5.                                                                        s5 s7                                                                                    Methods. Artifical Intelligence      124 (2000)

s4
s6            s8 s10                                 A traverse decomposition for Hcg
10. Gottlob, G., Hutle, M., Wotawa, F.: Combining Hypertree, Bicomp, And Hinge
s2       s5 s6         s6     s7        s9 s9 s10                         s9                                  s3        s1      starting from {s6, s9, s12}.
Decomposition. ECAI 02 (2002)

s1 s2 s3    s4       s6      s7 s7 s8 s8 s8 s                                               s14 s15 s16             s4                                                              11. Zheng, Y., Choueiry B.Y.: Cut-and-Traverse: A New Structural Decomposition Strategy
s11         s12 s s s s9 9 s9 s15                               s12                                                   Width of the join tree is 10.                     for Finite Constraint Satisfaction Problems. CSCLP 04 (2004).
s2    s4    s5 s s12             13 13 14 s
14 s9
s11 s12 s17
s5          12        s13    s14            s s      9 16
s11
This research is supported by CAREER Award #0133568
s11 s17              Applying hinge+ decomposition to Hcg.                                                                                                            September 8, 2004                        from the National Science Foundation.

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