Cut-and-Traverse: A new Structural Decomposition Method for CSPs

1
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_at_cse.unl.edu

• General principal
• 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
• Solve each sub-problem (all solutions), usually
  by a join operation.
• Apply directional arc-consistency to the
  constraint tree T.
• Find a solution for the CSP using backtrack-free
  search.
• Goal a decomposition technique that is efficient
  and minimizes width of tree

• Cut-and-Traverse decomposition has the following
  steps
• 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 C1, then
  traverse it from C1.
• If the tree node contains two cuts C1 and C2,
  then traverse it from C1 to C2.
• 3. Combine the traverse results.

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 s4, s5, s6, s11, s12 contains
3 cuts s4, s5, s6, s12, and s11.
A Cut-and-Traverse decomposition of Hcg. Cut
limit size is 2. Width of the join tree is 2.
Applying cut decomposition to Hcg.

1. All these decomposition methods can be performed
   in polynomial time.

Traverse Decomposition
Hinge decomposition O(VEk1) Cut
decomposition O(VEk1) Traverse
decompositionO(VE2) Cut-and-Traverse
decompositionO(VEk1) k is the limit size for
cuts

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

• We traverse a constraint hypergraph from a set F
  of hyperedges, until all the hyperedges are
  visited as follows
• Start from Fs, mark all hyperedges whose vertices
  contained in Fs as visited.
• Then traverse to Fss unvisited neighbors F1,
  mark all hyperedges whose vertices contained in
  F1 as visited.
• Then traverse to F1s unvisited neighbors F2,
  mark all hyperedges whose vertices contained in
  F2 as visited.
• Continuously traversing until all the hyperedges
  are visited.

1. Hinge decomposition strongly generalizes hinge
   decomposition.
2. Cut-and-traverse strongly generalizes cut
   decomposition.
3. Hypertree decomposition strongly generalizes
   hinge decomposition, traverse decomposition,
   Cut-and-Traverse decomposition.

A traverse decomposition for Hcg starting from
s1. Width of the join tree is 3.
A constraint hypergraph Hcg .

• Empirically evaluate and compare the new proposed
  decomposition methods on randomly generated
  constraint hypergraphs.
• Compare cut-and-traverse decomposition method
  with
• hinge decomposition tree clustering method, and
• hinge decomposition biconnected component
  hypertree decomposition.

• We traverse a constraint hypergraph from a set Fs
  of hyperedges to another set of hyperedges Fd as
  follows
• Start from Fs, mark all hyperedges whose vertices
  contained in Fs as visited.
• Then traverse to Fss unvisited neighbors and
  those hyperedges in Fd that has common vertices
  with Fs, we denote them as F1, mark all
  hyperedges whose vertices contained in F1 as
  visited.
• Then traverse to F1s unvisited neighbors and
  those hyperedges in Fd that has common vertices
  with F1, we denote them as F2, mark all
  hyperedges whose vertices contained in F2 as
  visited.
• Continuously traversing until traversing to Fd
  and all the hyperedges are visited.

• Given a constraint hypergraph H (V, E) where H
  is connected and E k1. We call a k-cut of H
  a set F of hyperedges that satisfies the
  following conditions
• F is a subset of E and F k, and
• The remaining constraint hypergraph H1, , Hq has
  at least 2 components.

References

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• Hinge decomposition of Hcg
• Hinge decomposition continuously finds 1-cut in
  in Hcg
• Width of the 1-hinge tree to the right is 12.

A traverse decomposition for Hcg starting from
s1, s2 to s9, s16. Width of the join tree is
3.
Applying hinge decomposition to Hcg.
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.
Hinge decomposition of Hcg

• After finding all the 1-cuts, we continuously
  find 2-cuts in Hcg.
• 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.

A traverse decomposition for Hcg starting from
s6, s9, s12. Width of the join tree is 10.
This research is supported by CAREER Award
0133568 from the National Science Foundation.
Applying hinge decomposition to Hcg.
September 8, 2004